CIVIL   ENG,    DEPT. 


OF  CALIFORNIA 
*€I»AET1V2ENT  QF  CIVIL 

•vW 


GEODETIC  SURVEYING 


-Ml  Book  &,  7m 


PUBLISHERS     OF     BOOKS      F  O  R_y 


Electrical  World  ^  Engineering  News  'Record 
Power  v  Engineering  and  Mining  Journal-Press 
Chemical  and  Metallurgical  Engineering 
Electric  Railway  Journal  v  Coal  Age 
American  Machinist  v  Ingenierifc  Internacional 
Electrical  Merchandising  v  BusTransportation 
Journal  of  Electricity  and  Western  Industry 
Industrial  Engineer 


GEODETIC    SURVEYING 


THE   ADJUSTMENT   OF    OBSERVATIONS 

(METHOD  OF  LEAST  SQUARES) 


BY 


EDWARD  L.  INGRAM,  C.E. 

Professor  of  Railroad  Engineering  and  Cfeodesy,  University  of  Pennsylvania 


FIRST  EDITION 
FOURTH  IMPRESSION 


McGRAW-HILL  BOOK  COMPANY,  INC. 
NEW  YORK:  370  SEVENTH  AVENUE 

LONDON:  6  &  8  BOUVERIE  ST.,  E.  C.  4 
1911 


Engineering 
Library 


COPYRIGHT,  1911,  BY  THE 
McGRAW-HILL  BOOK  COMPANY.  INC. 


PRINTED   IN   THE    UNITED    STATES    OF   AMERICA 


PREFACE 


AFTER  a  careful  examination  of  existing  books,  the  University 
of  Pennsylvania  has  failed  to  find  a  satisfactory  text  from  which  to 
teach  its  civil  engineering  students  the  fundamental  principles  of 
geodetic  surveying  and  the  adjustment  of  observations  as  it 
feels  they  should  be  taught  to  this  class  of  men.  A  canvass  of 
the  leading  colleges  of  the  country  has  shown  that  the  same  lack 
of  a  suitable  book  has  been  felt  by  many  other  institutions.  The 
present  volume  has  been  prepared  to  meet  this  apparent  need. 
No  attempt  has  been  made,  therefore,  to  treat  the  subject 
exhaustively  for  the  benefit  of  the  professional  geodesist,  but 
rather  to  build  up  a  book  containing  everything  that  can  be  con- 
sidered desirable  for  the  student  or  useful  to  the  practicing  civil 
engineer.  In  order  to  make  the  book  complete  for  such  engineers 
it  has  been  necessary  to  include  a  large  amount  of  matter  not 
desirable  or  suitable  for  class-room  work,  the  arrangement  of 
the  college  course  being  left  to  the  judgment  of  the  instructor. 

In  writing  the  book  in  two  parts  the  aim  has  been  to  make  each 
part  complete  in  itself,  so  that  either  part  may  be  read  intelligently 
without  having  read  the  other  part.  Those  who  wish  to  make 
a  study  of  geodetic  work  without  entering  into  involved  mathe- 
matical discussions,  will  find  a  complete  treatment  of  geodetic 
methods  and  the  rules  for  making  the  necessary  adjustments 
in  the  first  part  of  the  book.  Those  who  wish  to  become  familiar 
with  the  fundamental  principles  of  least  squares,  or  those  familiar 
with  geodetic  work  who  wish  to  understand  the  mathematical 
theory  on  which  the  rules  for  adjusting  observations  are  based, 
may  read  the  second  part  of  the  book  alone.  The  book  has 
been  written  with  the  intention,  however,  that  engineering  students 
shall  take  the  two  parts  in  succession. 


vi  PEEFACE 

In  the  first  part  of  the  book  the  initial  chapter  takes  up  the 
principles  of  triangulation  work  as  the  best  introduction  to  geodetic 
work  in  general.  Nothing  new  of  any  special  importance  is 
available  in  the  general  scheme  of  triangulation,  and  the  chapter 
is  written  as  briefly  and  logically  as  possible. 

The  second  chapter  treats  of  the  subject  of  base-line  measure- 
ment, including  measurements  with  base-bars,  steel  tapes,  invar 
tapes,  and  steel  and  brass  wires.  Special  care  has  been  taken 
to  have  such  constants  as  the  temperature  coefficient,  the  modulus 
of  elasticity,  and  the  specific  weight  correct  and  complete  for 
the  different  materials  involved.  The  mathematical  treatment 
of  the  corrections  required  in  base-line  work  has  been  made  as 
simple  as  possible,  avoiding  needless  transformations  of  mathemat- 
ical formulas  to  cover  unusual  methods  of  work. 

The  third  chapter  takes  up  the  subject  of  angle  measurement, 
and  is  intended  to  make  clear  the  most  approved  methods  of  using 
the  instruments  and  performing  the  actual  work  in  the  field. 
The  repeating  method  is  given  in  much  detail  on  account  of  the 
excellent  results  obtainable  by  this  method  with  the  ordinary 
engineer's  transit. 

The  fourth  chapter  includes  the  computations  and  adjust- 
ments required  in  triangulation  work,  and  is  intended  to  cover 
all  points  of  interest  to  the  civil  engineer. 

The  fifth  chapter  takes  up  the  subject  of  computing  the  geodetic 
positions  from  the  results  of  the  triangulation  work.  The  mathe- 
matical treatment  of  this  subject  is  so  difficult  that  the  formulas 
to  be  used  are  given  without  demonstration,  but  all  the  rules  and 
constants  are  given  that  the  engineer  will  ever  require. 

The  sixth  chapter  is  devoted  to  geodetic  leveling,  and  contains 
the  familiar  knowledge  on  this  subject  arranged  as  briefly  as  is 
consistent  with  clearness  and  completeness. 

'  The  seventh  chapter  is  devoted  to  astronomical  determina- 
tions, giving  in  detail  such  work  as  falls  within  the  province  of  the 
engineer,  and  in  outline  such  general  information  as  the  educated 
engineer  should  possess,  but  which  is  seldom  found  in  engineering 
text-books.  The  number  of  methods  for  making  astronomical 
determinations  is  almost  without  limit,  but  the  older  and  well- 
tried  methods  are  here  retained  as  best  adapted  to  the  needs  of 
the  engineer. 

The  eighth  chapter  considers  the  principal  methods  of  map 


PREFACE  vii 

projection,  and  differs  from  the  treatment  found  in  other  books 
chiefly  by  including  the  formulas  which  alone  make  it  possible 
to  use  the  different  methods. 

Chapters  IX  to  XVI  form  the  second  part  of  the  book,  devoted 
to  the  development  of  the  Method  of  Least  Squares  and  its  applica- 
tion to  the  adjustment  of  observations. 

Chapter  IX  includes  the  necessary  classification  of  values, 
quantities  and  errors,  and  also  the  laws  of  chance  on  which  the 
theory  of  errors  is  founded.  This  is  followed  in  Chapter  X  by 
the  development  of  the  mathematical  theory  of  errors,  which  is 
the  fundamental  basis  from  which  all  the  rules  for  adjustments 
are  derived. 

Chapter  XI  develops  the  mathematical  methods  for  obtaining 
the  most  probable  values  of  independent  quantities  in  general 
from  their  observed  values,  and  Chapter  XII  extends  the  methods 
so  as  to  include  conditioned  and  computed  quantities. 

Chapter  XIII  explains  the  meaning  of  and  methods  of  obtaining 
the  probable  error  for  both  observed  and  computed  quantities. 
The  derivation  of  the  necessary  formulas  is  considered  too 
abstruse  for  the  average  student,  and  these  formulas  are  given 
without  demonstration. 

Chapters  XIV,  XV,  and  XVI,  deal  respectively  with  the 
application  of  the  theory  of  least  squares  to  the  various  condi- 
tions met  with  and  adjustments  required  in  angle  work,  base- 
line work,  and  level  work,  covering  all  cases  likely  to  be  of  interest 
to  the  civil  engineer. 

In  the  preparation  of  the  text  the  following  points  have  been 
kept  constantly  in  view :  to  bring  the  book  up  to  date ;  to  make  the 
treatment  of  each  subject  as  clear  and  concise  as  possible;  to 
use  the  same  symbols  throughout  the  book  for  the  same  meaning, 
adopting  ,the  symbols  having  the  most  general  acceptance;  to 
define  each  symbol  in  a  formula  where  the  formula  is  developed, 
so  that  the  user  of  the  formula  is  never  required  to  hunt  for  the 
meaning  of  its  terms;  to  give  for  every  formula  the  unit  in  which 
each  symbol  is  to  be  taken;  to  clear  up  any  doubt  as  to  what 
algebraic  sign  is  to  be  given  to  a  symbol  in  a  formula,  as  the  sign 
required  in  a  geodetic  formula  is  not  infrequently  the  opposite 
of  what  would  naturally  be  supposed;  to  make  perfectly  rigid 
such  demonstrations  as  are  given;  where  demonstrations  are 
not  given  to  state  where  they  may  be  found;  to  give  the  best 


viii  PREFACE 

obtainable  values  for  all  constants  required  in  geodetic  work; 
and  to  state  the  accuracy  attainable  with  different  instruments 
and  methods,  so  that  a  proper  choice  may  be  made.  Attention 
is  called  to  the  very  large  number  of  illustrative  examples  that 
are  given,  and  which  are  worked  out  in  detail  so  that  every 
process  may  be  thoroughly  understood. 

E.  L.  I. 
PHILADELPHIA,  PA.,  December,  1911. 


TABLE  OF  CONTENTS 


INTRODUCTION 

ART.  PAGE 

1.  GEODESY  DEFINED. 1 

2.  IMPORTANCE  OF  GEODETIC  WORK 1 

3.  GEODETIC  WORK  IN  THE  UNITED  STATES 1 

4.  HISTORICAL  NOTES 1 

5.  SCOPE  OF  GEODESY 2 

6.  GEODETIC  SURVEYING 2 

7.  ADJUSTMENT  OF  OBSERVATIONS 3 


PART  I 
GEODETIC  SURVEYING 


CHAPTER  I 

PRINCIPLES  OF  TRIANGULATION 

8.  GENERAL  SCHEME '4 

9.  GEOMETRICAL  CONDITIONS 5 

10.  SPECIAL  CASES 8 

11.  CLASSIFICATION  OF  TRIANGULATION  SYSTEMS 9 

12.  SELECTION  OF  STATIONS 10 

13.  RECONNOISSANCE 10 

14.  CURVATURE  AND  REFRACTION 12 

15.  INTERVISIBILITY  OF  STATIONS 14 

Example 15 

16.  HEIGHT  OF  STATIONS 17 

Example 17 

17.  STATION  MARKS 17 

18.  OBSERVING  STATIONS  AND  TOWERS 18 

ix 


x  TABLE  OF  CONTENTS 

ART.  PAGE 

19.  STATION  SIGNALS  OR  TARGETS 18 

Board  Signals 20 

Pole  Signals 20 

Heliotropes 21 

Night  Signals 23 

CHAPTER  II 
BASE-LINE  MEASUREMENT 

20.  GENERAL  SCHEME 24 

21.  BASE-BARS  AND  THEIR  USE 24 

22.  STEEL  TAPES  AND  THEIR  USE 30 

23.  INVAR  TAPES 32 

24.  MEASUREMENTS  WITH  STEEL  AND  BRASS  WIRES 32 

25.  STANDARDIZING  BARS  AND  TAPES 33 

26.  CORRECTIONS  REQUIRED  IN  BASE-LINE  WORK 33 

27.  CORRECTION  FOR  ABSOLUTE  LENGTH 36 

28.  CORRECTION  FOR  TEMPERATURE 36 

29.  CORRECTION  FOR  PULL 38 

30.  CORRECTION  FOR  SAG 39 

31.  CORRECTION  FOR  HORIZONTAL  ALIGNMENT 40 

32.  CORRECTION  FOR  VERTICAL  ALIGNMENT 42 

33.  REDUCTION  TO  MEAN  SEA  LEVEL 43 

34.  COMPUTING  GAPS  IN  BASE  LINES 44 

35.  ACCURACY  OF  BASE-LINE  MEASUREMENTS 45 

Example 46 

CHAPTER  III 
MEASUREMENT  OF  ANGLES 

36.  GENERAL  CONDITIONS 47 

37.  INSTRUMENTS  FOR  ANGULAR  MEASUREMENTS 47 

38.  THE  REPEATING  INSTRUMENT  AND  ITS  USE 52 

39.  FIRST  METHOD  WITH  REPEATING  INSTRUMENT 52 

40.  SECOND  METHOD.  WITH  REPEATING  INSTRUMENT 53 

40a.  Reducing  the  Notes 56 

406.  Illustrative  Example 57 

40c.  Additional  Instructions 58 

41.  ADJUSTMENTS  OF  THE  REPEATING  INSTRUMENT 59 

42.  THE  DIRECTION  INSTRUMENT  AND  ITS  USE 60 

43.  FIRST  METHOD  WITH  DIRECTION  INSTRUMENT 61 

44.  SECOND  METHOD  WITH  DIRECTION  INSTRUMENT 64 

45.  THE  MICROMETER  MICROSCOPES 65 

46.  READING  THE  MICROMETERS 67 

46a.  Run  of  the  Micrometer 68 

Example 72 


TABLE  OF  CONTENTS  xi 

ART.  PAGE 

47.  ADJUSTMENTS  OF  THE  DIRECTION  INSTRUMENT . 71 

48.  REDUCTION  TO  CENTER 75 

49.  ECCENTRICITY  OF  SIGNAL 78 

50.  ACCURACY  OF  ANGLE  MEASUREMENTS 78 

Example 79 


CHAPTER  IV 
TRIANGULATION  ADJUSTMENTS  AND  COMPUTATIONS 

51.  ADJUSTMENTS 81 

52.  THEORY  OF  WEIGHTS 81 

53.  LAWS  OF  WEIGHTS 82 

Examples 82 

54.  STATION  ADJUSTMENT 84 

Examples 84 

55.  FIGURE  ADJUSTMENT ' 87 

56.  SPHERICAL  EXCESS 88 

57.  TRIANGLE  ADJUSTMENT 89 

58.  THE  GEODETIC  QUADRILATERAL 90 

59.  APPROXIMATE  ADJUSTMENT  OF  A  QUADRILATERAL 92 

Example 95 

60.  RIGOROUS  ADJUSTMENT  OF  A  QUADRILATERAL 96 

Example 101 

61.  WEIGHTED  ADJUSTMENTS  AND  LARGER  SYSTEMS 100 

62.  COMPUTING  THE  LINES  OF  THE  SYSTEM 102 

63.  ACCURACY  OF  TRIANGULATION  WORK.  .                                               .  102 


CHAPTER  V 
COMPUTING  THE  GEODETIC  POSITIONS 

64.  THE  PROBLEM 103 

65.  THE  FIGURE  OF  THE  EARTH 104 

66.  THE  PRECISE  FIGURE 105 

67.  THE  PRACTICAL  FIGURE 106 

68.  GEOMETRICAL  CONSIDERATIONS 106 

69.  ANALYTICAL  CONSIDERATIONS 109 

70.  CONVERGENCE  OF  THE  MERIDIANS Ill 

71.  THE  PUISSANT  SOLUTION 113 

72.  THE  CLARKE  SOLUTION 116 

73.  THE  INVERSE  PROBLEM 118 

74.  LOCATING  A  PARALLEL  OF  LATITUDE 120 

75.  DEVIATION  OF  THE  PLUMB  LINE .124 


xii  TABLE  OF  CONTENTS 


CHAPTER  VI 
GEODETIC   LEVELING 

ART.  PAGE 

76.  PRINCIPLES  AND  METHODS 125 

77.  DETERMINATION  OF  MEAN  SEA  LEVEL 125 

A.  BAROMETRIC  LEVELING 

78.  INSTRUMENTS  AND  METHODS 126 

79.  THE  COMPUTATIONS 128 

Example 128 

80.  ACCURACY  OF  BAROMETRIC  WORK 129 

B.  TRIGONOMETRIC  LEVELING 

81.  INSTRUMENTS  AND  METHODS 130 

82.  BY  THE  SEA  HORIZON  METHOD 131 

83.  BY  AN  OBSERVATION  AT  ONE  STATION 133 

84.  BY  RECIPROCAL  OBSERVATIONS 136 

85.  COEFFICIENT  OF  REFRACTION 138 

86.  ACCURACY  OF  TRIGONOMETRIC  LEVELING 139 

C.  PRECISE  SPIRIT  LEVELING 

87.  INSTRUMENTAL  FEATURES : 139 

88.  GENERAL  FIELD  METHODS 143 

89.  THE  EUROPEAN  LEVEL 145 

89a.  Constants  of  European  Level 146 

Examples 147 

89&.  Adjustments  of  European  Level 150 

Example 152 

89c.  Use  of  European  Level 152 

Example 154 

90.  THE  COAST  SURVEY  LEVEL 153 

90a.  Constants  of  Coast  Survey  Level 155 

906.  Adjustments  of  Coast  Survey  Level 155 

90c.  Use  of  Coast  Survey  Level 156 

Example 157 

91.  RODS  AND  TURNING  POINTS 158 

92.  ADJUSTMENT  OF  LEVEL  WORK 160 

Example 160 

93.  ACCURACY  OF  PRECISE  SPIRIT  LEVELING 161 


TABLE  OF  CONTENTS  xiii 


-CHAPTER  VII 
ASTRONOMICAL  DETERMINATIONS 

ART.  PAGE 

94.  GENERAL  CONSIDERATIONS.  .  .   163 


TIME 

95.  GENERAL  PRINCIPLES 164 

96.  MEAN  SOLAR  TIME 164 

96a.  Standard  Time 165 

966.  To   Change   Standard  Time  to  Local  Mean  Time  and  vice 

versa 165 

Examples 168 

97.  SIDEREAL  TIME 168 

98.  To  CHANGE  A  SIDEREAL  TO  A  MEAN  TIME  INTERVAL  AND  vice 

versa 169 

99.  To  CHANGE  LOCAL  MEAN  TIME  OR  STANDARD  TIME  TO  SIDEREAL  .  .  169 

Example 169 

100.  To  CHANGE  SIDEREAL  TO  LOCAL  MEAN  TIME  OR  STANDARD  TIME  .  .  170 

Example 170 

101.  TIME  BY  SINGLE  ALTITUDES 171 

lOla.  Making  the  Observation 172 

1016.  The  Computation 173 

Example 175 

102.  TIME  BY  EQUAL  ALTITUDES 176 

102a.  Making  the  Observation 176 

1026.  The  Computation 177 

Example 180 

103.  TIME  BY  SUN  AND  STAR  TRANSITS 181 

103a.  Sun  Transits  with  Engineering  Instruments 181 

1036.  Star  Transits  with  Engineering  Instruments 182 

103c.  Star  Transits  with  Astronomical  Instruments 183 

104.  CHOICE  OF  METHODS 184 

105.  TIME  DETERMINATIONS  AT  SEA  .  .                                                        .  184 


LATITUDE 

106.  GENERAL  PRINCIPLES 186 

107.  LATITUDE  FROM  OBSERVATIONS  ON  THE  SUN  AT  APPARENT  NOON  ....  188 

108.  LATITUDE  BY  CULMINATION  OF  CIRCUMPOLAR  STARS 190 

109.  LATITUDE  BY  PRIME-VERTICAL  TRANSITS 192 

110.  LATITUDE  WITH  THE  ZENITH  TELESCOPE 193 

111.  LATITUDE  DETERMINATIONS  AT  SEA 196 

112.  PERIODIC  CHANGES  IN  LATITUDE.  .  .  196 


xiv  TABLE  OF  CONTENTS 

LONGITUDE 

ART.  PAGE 

113.  GENERAL  PRINCIPLES 197 

114.  DIFFERENCE  OF  LONGITUDE  BY  SPECIAL  METHODS 198 

By  Special  Phenomena 198 

By  Flash  Signals 198 

115.  LONGITUDE  BY  LUNAR  OBSERVATIONS 198 

By  Lunar  Distances 199 

By  Lunar  Culminations 199 

By  Lunar  Occultations 199 

116.  DIFFERENCE  OF  LONGITUDE  BY  TRANSPORTATION  OF  CHRONOMETERS.  199 

117.  DIFFERENCE  OF  LONGITUDE  BY  TELEGRAPH 200 

By  Standard  Time  Signals • 201 

By  Star  Signals 201 

By  Arbitrary  Signals 202 

118.  LONGITUDE  DETERMINATIONS  AT  SEA 203 

119.  PERIODIC  CHANGES  IN  LONGITUDE 203 

AZIMUTH 

120.  GENERAL  PRINCIPLES ; 203 

121.  THE  AZIMUTH  MARK 204 

122.  AZIMUTH  BY  SUN  OR  STAR  ALTITUDES 205 

122a.  Making  the  Observation 205 

1226.  The  Computation 206 

123.  AZIMUTH  FROM  OBSERVATIONS  ON  CIRCUMPOLAR  STARS 207 

123a.  Fundamental  Formulas 208 

1236.  Approximate  Determinations 213 

123c.  The  Direction  Method 215 

Example 216 

123d.  The  Repeating  Method 218 

Example 219 

123e.  The  Micrometric  Method 221 

Example 223 

124.  AZIMUTH  DETERMINATIONS  AT  SEA 225 

125.  PERIODIC  CHANGES  IN  AZIMUTH , 226 

CHAPTER  VIII 
GEODETIC  MAP  DRAWING 

126.  GENERAL  CONSIDERATIONS 227 

127.  CYLINDRICAL  PROJECTIONS 229 

127a.  Simple  Cylindrical  Projection 229 

1276.  Rectangular  Cylindrical  Projection 231 

127c.  Mercator's  Cylindrical  Projection 231 

128.  TRAPEZOIDAL  PROJECTION  .  .  234 


TABLE  OF  CONTENTS  xv 


ART. 


129.  CONICAL  PROJECTIONS  ..........  .  ............................  234 

129a.  Simple  Conic  Projection  .  .*?  ............................  235 

1296.  Mercator's  Conic  Projection  ............................  236 

129c.  Bonne's  Conic  Projection  ..............................  238 

129d.  Polyconic  Projection  ..................................  239 


PART  II 

ADJUSTMENT  OF  OBSERVATIONS  BY  THE 
METHOD  OF  LEAST  SQUARES 


CHAPTER  IX 
DEFINITIONS   AND   PRINCIPLES 

130.  GENERAL  CONSIDERATIONS 241 

131.  CLASSIFICATION  OF  QUANTITIES 241 

132.  CLASSIFICATION  OF  VALUES 242 

133.  OBSERVED  VALUES  AND  WEIGHTS 243 

134.  MOST  PROBABLE  VALUES  AND  WEIGHTS 243 

135.  TRUE  AND  RESIDUAL  ERRORS 245 

136.  SOURCES  OF  ERROR 247 

137.  NATURE  OF  ACCIDENTAL  ERRORS 247 

138.  THE  LAWS  OF  CHANCE 248 

139.  SIMPLE  EVENTS 248 

140.  COMPOUND  EVENTS 249 

141.  CONCURRENT  EVENTS 249 

142.  MISAPPLICATION  OF  THE  LAWS  OF  CHANCE 250 

CHAPTER  X 
THE  THEORY  OF  ERRORS 

143.  THE  LAWS  OF  ACCIDENTAL  ERROR 252 

144.  GRAPHICAL  REPRESENTATION  OF  THE  LAWS  OF  ERROR 253 

145.  THE  Two  TYPES  OF  ERROR 254 

146.  THE  FACILITY  OF  ERROR 255 

147.  THE  PROBABILITY  OF  ERROR 256 

148.  THE  LAW  OF  THE  FACILITY  OF  ERROR 257 

149.  FORM  OF  THE  PROBABILITY  EQUATION 257 

150.  GENERAL  EQUATION  OF  THE  PROBABILITY  CURVE 260 

151.  THE  VALUE  OF  THE  PRECISION  FACTOR 262 

152.  COMPARISON  OF  THEORY  AND  EXPERIENCE  .  .  .  264 


xvi  TABLE  OF  CONTENTS 


CHAPTER  XI 
MOST  PROBABLE  VALUES  OF  INDEPENDENT  QUANTITIES 

ART.  PAGE 

153.  GENERAL  CONSIDERATIONS 266 

154.  FUNDAMENTAL  PRINCIPLE  OF  LEAST  SQUARES 266 

155.  DIRECT  OBSERVATIONS  OF  EQUAL  WEIGHT 267 

Example 268 

156.  GENERAL  PRINCIPLE  OF  LEAST  SQUARES 268 

157.  DIRECT  OBSERVATIONS  OF  UNEQUAL  WEIGHT 270 

Example 271 

158.  INDIRECT  OBSERVATIONS 271 

159.  INDIRECT  OBSERVATIONS  OF  EQUAL  WEIGHT  ON  INDEPENDENT 

QUANTITIES 273 

Examples 274 

160.  INDIRECT  OBSERVATIONS  OF  UNEQUAL  WEIGHT  ON  INDEPENDENT 

QUANTITIES 276 

Examples 277 

161.  REDUCTION  OF  WEIGHTED  OBSERVATIONS  TO  EQUIVALENT  OBSER- 

VATIONS OF  UNIT  WEIGHT 278 

Examples 279 

162.  LAW  OF  THE  COEFFICIENTS  IN  NORMAL  EQUATIONS 280 

Example 281 

163.  REDUCED  OBSERVATION  EQUATIONS. 281 

Examples 282 

CHAPTER  XII 

MOST  PROBABLE  VALUES  OF  CONDITIONED  AND  COMPUTED 

QUANTITIES 

164.  CONDITIONAL  EQUATIONS 284 

165.  AVOIDANCE  OF  CONDITIONAL  EQUATIONS 285 

Examples 286 

166.  ELIMINATION  OF  CONDITIONAL  EQUATIONS 288 

Example 289 

167.  METHOD  OF  CORRELATIVES 290 

Example 295 

168.  MOST  PROBABLE  VALUES  OF  COMPUTED  QUANTITIES 296 

CHAPTER  XIII 

PROBABLE  ERRORS  OF  OBSERVED  AND  COMPUTED 
QUANTITIES 

A.  OF  OBSERVED  QUANTITIES 

169.  GENERAL  CONSIDERATIONS 297 

170.  FUNDAMENTAL  MEANING  OF  THE  PROBABLE  ERROR 297 


TABLE  OF  CONTENTS  xvii 

ART.  ,  PAGE 

171.  GRAPHICAL  REPRESENTATION  OF  £HE  PROBABLE  ERROR 298 

172.  GENERAL  VALUE  OF  THE  PROBABLE  ERROR 299 

173.  DIRECT  OBSERVATIONS  OF  EQUAL  WEIGHT 300 

Example 300 

174.  DIRECT  OBSERVATIONS  OF  UNEQUAL  WEIGHT 301 

Example 302 

175.  INDIRECT  OBSERVATIONS  ON  INDEPENDENT  QUANTITIES 302 

Example 303 

176.  INDIRECT  OBSERVATIONS  INVOLVING  CONDITIONAL  EQUATIONS 304 

177.  OTHER  MEASURES  OF  PRECISION 304 

B.  OF  COMPUTED  QUANTITIES 

178.  TYPICAL  CASES 306 

179.  THE   COMPUTED  QUANTITY  is  THE  SUM  OR  DIFFERENCE   OF  AN 

OBSERVED  QUANTITY  AND  A  CONSTANT 306 

Example 307 

180    THE    COMPUTED    QUANTITY    is    OBTAINED    FROM    AN    OBSERVED 

QUANTITY  BY  THE  USE  OF  A  CONSTANT  FACTOR 307 

Example 308 

181.  THE  COMPUTED  QUANTITY  is  ANY  FUNCTION  OF  A  SINGLE  OB- 

SERVED QUANTITY 308 

Example 308 

182.  THE  COMPUTED  QUANTITY  is  THE  ALGEBRAIC  SUM  OF  Two  OR 

MORE  INDEPENDENTLY  OBSERVED  QUANTITIES 308 

Examples 309 

183.  THE  COMPUTED  QUANTITY  is  ANY  FUNCTION  OF  Two  OR  MORE 

INDEPENDENTLY  OBSERVED  QUANTITIES 310 

Examples 310 

CHAPTER  XIV 
APPLICATION  TO  ANGULAR  MEASUREMENTS 

184.  GENERAL  CONSIDERATIONS 312 

SINGLE  ANGLE  ADJUSTMENT 

185.  THE  CASE  OF  EQUAL  WEIGHTS 312 

Example 312 

186.  THE  CASE  OF  UNEQUAL  WEIGHTS 313 

Example 313 

STATION  ADJUSTMENT 

187.  GENERAL  CONSIDERATIONS 313 

188.  CLOSING  THE  HORIZON  WITH  ANGLES  OF  EQUAL  WEIGHT 313 

Example 315 


xviii  TABLE  OF  CONTENTS 

ART.  PAGE 

189.  CLOSING  THE  HORIZON  WITH  ANGLES  OF  UNEQUAL    WEIGHT 315 

Example 317 

190.  SIMPLE  SUMMATION  ADJUSTMENTS 317 

Examples 318 

191.  THE  GENERAL  CASE : 319 

Examples 320 

FIGURE  ADJUSTMENT 

192.  GENERAL  CONSIDERATIONS 321 

193.  TRIANGLE  ADJUSTMENT  WITH  ANGLES  OF  EQUAL  WEIGHT 322 

Example 323 

194.  TRIANGLE  ADJUSTMENT  WITH  ANGLES  OF  UNEQUAL  WEIGHT 323 

Example 324 

195.  Two  CONNECTED  TRIANGLES 325 

Example 325 

196.  QUADRILATERAL  ADJUSTMENT 326 

Example 330 

197.  OTHER  CASES  OF  FIGURE  ADJUSTMENT 329 

Examples 331 


CHAPTER  XV 
APPLICATION  TO  BASE-LINE  WORK 

198.  UNWEIGHTED  MEASUREMENTS 333 

Example 333 

199.  WEIGHTED  MEASUREMENTS 333 

Example 334 

200.  DUPLICATE  LINES 334 

Example 335 

201.  SECTIONAL  LINES 335 

Example 336 

202.  GENERAL  LAW  OF  THE  PROBABLE  ERRORS 336 

Example 337 

203.  THE  LAW  OF  RELATIVE  WEIGHT 337 

204.  PROBABLE  ERROR  OF  A  LINE  OF  UNIT  LENGTH 338 

205.  DETERMINATION  OF  THE   NUMERICAL  VALUE   OF  THE   PROBABLE 

ERROR  OF  A  LINE  OF  UNIT  LENGTH 339 

Example 341 

206.  THE  UNCERTAINTY  OF  A  BASE  LINE 342 

Examples 343 


TABLE  OF  CONTENTS  xix 


CHAPTER  XVI 

p 

APPLICATION  TO  LEVEL  WORK 

ART.  PAGE 

207.  UNWEIGHTED  MEASUREMENTS 344 

Example 344 

208.  WEIGHTED  MEASUREMENTJ 345 

Example 345 

209.  DUPLICATE  LINES 346 

Example 346 

210.  SECTIONAL  LINES 347 

Example 347 

211.  GENERAL  LAW  OF  THE  PROBABLE  ERRORS 347 

Example 348 

212.  THE  LAW  OF  RELATIVE  WEIGHT 348 

213.  PROBABLE  ERROR  OF  A  LINE  OF  UNIT  LENGTH 348 

214.  DETERMINATION  OF  THE   NUMERICAL  VALUE   OF  THE   PROBABLE 

ERROR  OF  A  LINE  OF  UNIT  LENGTH 349 

Example 350 

215.  MULTIPLE  LINES 350 

Example 351 

216.  LEVEL  NETS 352 

Example 353 

217.  INTERMEDIATE  POINTS 355 

Example 356 

218.  CLOSED  CIRCUITS 357 

Example 358 

219.  BRANCH  LINES,  CIRCUITS,  AND  NETS 359 


FULL-PAGE  PLATES 

EXAMPLE  OF  A  TRIANGULATION  SYSTEM 6 

EXAMPLE  OF  A  TOWER  STATION 19 

EIMBECK  DUPLEX  BASE-BAR 28 

CONTACT  SLIDES,  EIMBECK  DUPLEX  BASE-BAR 29 

REPEATING  INSTRUMENT 49 

DIRECTION  INSTRUMENT 50 

ALTAZIMUTH  INSTRUMENT 51 

REDUCTION  TO  CENTER 77 

LOCATION  OF  A  BOUNDARY  LINE , . . .  123 

EUROPEAN  TYPE  OF  PRECISE  LEVEL 141 

COAST  SURVEY  PRECISE  LEVEL 142 

MOLITOR'S  PRECISE  LEVEL  ROD  AND  JOHNSON'S  FOOT-PIN 159 

CELESTIAL  SPHERE 166 

PORTABLE  ASTRONOMICAL  TRANSIT 185 

MAP  OF  ClRCUMPOLAR  STARS 191 

ZENITH  TELESCOPE .  195 


xx  TABLE  OF  CONTENTS 


TABLES 

PAGE 

I.  CURVATURE  AND  REFRACTION  IN  ELEVATION 363 

II.  LOGARITHMS  OF  THE  PUISSANT  FACTORS 364 

III.  BAROMETRIC  ELEVATIONS 366 

IV.  CORRECTION    COEFFICIENTS    TO    BAROMETRIC    ELEVATIONS    FOR 

TEMPERATURE  (FAHRENHEIT)  AND  HUMIDITY 368 

V.  LOGARITHMS  OF  RADIUS  OF  CURVATURE  (METRIC) 369 

VI.  LOGARITHMS  OF  RADIUS  OF  CURVATURE  (FEET) 370 

VII.  CORRECTIONS    FOR    CURVATURE    AND    REFRACTION    IN    PRECISE 

SPIRIT  LEVELING 370 

VIII.  MEAN  ANGULAR  REFRACTION 371 

IX.  ELEMENTS  OF  MAP  PROJECTIONS 372 

X.  CONSTANTS  AND  THEIR  LOGARITHMS 373 

BIBLIOGRAPHY 

REFERENCES  ON  GEODETIC  SURVEYING 374 

REFERENCES  ON  METHOD  OF  LEAST  SQUARES  .  .  .   375 


GEODETIC  SURVEYING 

AND 

THE    ADJUSTMENT    OF    OBSERVATIONS 

(METHOD    OF    LEAST    SQUARES) 


INTRODUCTION 

1.  Geodesy  is  that  branch  of  science  which  treats  of  making 
extended   measurements   on   the   surface   of   the   earth,   and   of 
related  problems.     Primarily  the  object  of  such  work  is  to  furnish 
precise  locations  for  the  controlling  points  of  extensive  surveys. 
The  determination  of  the  figure  and  dimensions  of  the  earth, 
however,  is  also  a  fundamental  object. 

2.  The    Importance  of    Geodetic  Work  is  recognized  by  all 
civilized  nations,  each  of  which  maintains  an  extensive  organi- 
zation for  this  purpose.     The  knowledge  thus  gained  of  the  earth 
and  its  surface  has  been  of  great  benefit  to  humanity.     In  further- 
ance of  this  object  an  International  Geodetic  Association  has  been 
formed  (1886),  and  includes  the  United  States  (1889)  in  its  mem- 
bership. 

3.  Geodetic  Work  in  the  United  States  is  carried  on  mainly  by 
the  United  States  Coast  and  Geodetic  Survey,  a  branch  of  the  De- 
partment of  Commerce  and  Labor.     The  valuable  papers  on  geo- 
detic work  published  by  this  department  may  be  obtained  free  of 
charge  by  addressing  the  "  Superintendent  United  States  Coast 
and  Geodetic  Survey,  Washington,  D.  C." 

4.  History.     Plane    surveying    dates    from    about    the    year 
2000   B.C.     Geodesy  literally  began  about  230  B.C.,  in  the  time  of 
Erastosthenes  and  the  famous  school  of  Alexandria,   at  which 
time  very  fair  results  were  secured  in  the  effort  to  determine  the 


2  GEODETIC  SURVEYING 

shape  and  size  of  the  earth.  Modern  geodesy  practically  began 
in  the  seventeenth  century  in  the  time  of  Newton,  owing  to 
disputes  concerning  the  shape  of  the  earth  and  the  flattening  of 
the  poles.  (See  Chapter  V  for  further  treatment  of  this  subject.) 

5.  The  Scope  of  Geodesy  originally  involved  only  the  shape 
of  the  earth  and  its  dimensions.     Modern  geodesy  covers  many 
topics,  the  principal  ones  being  about  as  follows : 

Leveling  (on  land) ; 
Soundings  (oceans,  lakes,  rivers); 
Mean  Sea  Level; 
Triangulation  ; 
Time;    ,    . 

'LatAuide  (by  observation) ; 
Longitude  (by  observation) ; 
Azimuth  (by  observation) ; 
Computation  of  Geodetic  Positions  (latitude,  longitude,  and 

azimuth  by  computation) ; 
Problems  of  Location; 
Figure  and  Dimensions  of  the  Earth; 
Configuration  of  the  Earth; 
Map  Projection; 
Gravity ; 

Terrestrial  Magnetism; 
Deviation  of  the  Plumb  Line; 
Tides  and  Tidal  Phenomena; 
Ocean  Currents; 
Meteorology. 

6.  Geodetic    Surveying.     This   class   of   surveying   is   distin- 
guished from  plane  surveying  by  the  fact  that  it  takes  account 
of  the  curvature  of  the  earth,  usually  necessitated  by  the  large 
distances  or  areas  covered.     Work  of  this  character  requires  the 
utmost  refinement  of  methods  and  instruments, 

1st,  Because  allowing  for  the  curvature  of  the  earth  is  in 

itself  a  refinement; 
2nd,  Because    small    measurements    have    to    be    greatly 

expanded ; 

3rd,  Because  the  magnitude  of  the  work  involves  an  accumu- 
lation of  errors. 

The  fundamental  operations  of  geodetic  surveying  are  Triangu- 
lation and  Precise  Leveling.     These  in  turn  require  the  deter- 


INTRODUCTION  3 

mination  of  time,  latitude,  longitude,  and  azimuth;  the  deter- 
mination of  mean  sea  Jevel ;  anid  a  knowledge  of  the  figure  and 
dimensions  of  the  earth.  The  first  part  of  this  book  covers 
such  points  on  these  subjects  as  are  likely  to  interest  the  civil 
engineer. 

7.  The  Adjustment  of  Observations.  All  measurements  are 
subject  to  more  or  less  unknown  and  unavoidable  sources  of 
error.  Repeated  measurements  of  the  same  quantity  can  not 
be  made  to  agree  precisely  by  any  refinement  of  methods  or 
instruments.  Measurements  made  on  different  parts  of  the  same 
figure  do  not  give  results  that  are  absolutely  consistent  with  the 
rigid  geometrical  requirements  of  the  case.  Some  method  of 
adjustment  is  therefore  necessary  in  order  that  these  discrepan- 
cies may  be  removed.  Obviously  that  method  of  adjustment 
will  be  the  most  satisfactory  which  assigns  the  most  probable 
values  to  the  unknown  quantities  in  view  of  all  the  measurements 
that  have  been  taken  and  the  conditions  which  must  be  satisfied. 
Such  adjustments  are  now  universally  made  by  the  Method  of 
Least  Squares.  The  application  of  this  method  to  the  elementary 
problems  of  geodetic  work  forms  the  subject-matter  of  the  second 
part  of  this  book. 


PART  I 
GEODETIC   SURVEYING 


CHAPTER  1 
PRINCIPLES   OF  TRIANGULATION 

8.  General  Scheme.  The  word  triangulalion,  as  used  in 
geodetic  surveying,  includes  all  those  operations  required  to 
determine  either  the  relative  or  the  absolute  positions  of  different 
points  on  the  surface  of  the  earth,  when  such  operations  are 
based  on  the  properties  of  plane  and  spherical  triangles.  By  the 
relative  position  of  a  point  is  meant  its  location  with  reference 
to  one  or  more  other  points  in  terms  of  angles  or  distance  as  may 
be  necessary.  In  geodetic  work  distances  are  usually  expressed 
in  meters,  and  are  always  reduced  to  mean  sea  level,  as  explained 
later  on.  By  the  absolute  position  of  a  point  is  meant  its  loca- 
tion by  latitude  and  longitude.  Strictly  speaking  the  absolute 
position  of  a  point  also  includes  its  elevation  above  mean  sea 
level,  but  if  this  is  desired  it  forms  a  special  piece  of  work,  and 
comes  under  the  head  of  leveling.  Directions  are  either  relative 
or  absolute.  The  relative  directions  of  the  lines  of  a  survey  are 
shown  by  the  measured  or  computed  angles.  The  absolute 
direction  of  a  line  is  given  by  its  azimuth,  which  is  the  angle  it 
makes  with  a  meridian  through  either  of  its  ends,  counting  clock- 
wise from  the  south  point  and  continuously  up  to  360°.  For 
reasons  which  will  appear  later  the  azimuth  of  a  line  must  always 
be  stated  in  a  way  that  clearly  shows  which  end  it  refers  to. 

In  the  actual  field  work  of  the  triangulation  suitable  points, 
called  stations,  are  selected  and  definitely  marked  throughout 
the  area  to  be  covered,  the  selection  of  these  stations  depending 
on  the  character  of  the  country  and  the  object  of  the  survey. 

4 


PRINCIPLES  OF  TRIANGULATION  5 

The  stations  thus  established  are  regarded  as  forming  the  vertices 
of  a  set  of  mutually  cpnnected^triangles  (overlapping  or  not,  as 
the  case  may  be),  the  complete  figure  being  called  a  triangula- 
tion  system.  At  least  one  side  and  all  the  angles  in  the  triangula- 
tion  system  are  directly  measured,  using  the  utmost  care.  All 
the  remaining  sides  are  obtained  by  computation  of  the  successive 
triangles,  which  (corrected  for  spherical  excess,  if  necessary) 
are  treated  as  plane  triangles.  The  line  which  is  actually  measured 
is  called  the  base  line.  It  is  common  to  measure  an  additional 
line  near  the  close  of  the  work,  this  line  being  connected  with 
the  triangulation  system  so  that  its  length  may  also  be  obtained 
by  calculation.  Such  a  line  is  called  a  check  base,  forming  an 
excellent  check  on  both  the  field  work  and  the  computations  of  the 
whole  survey.  In  work  of  large  extent  intermediate  bases  or  check 
bases  are  often  introduced.  Lines  which  are  actually  measured  on 
the  ground  are  always  reduced  to  mean  sea  level  before  any  further 
use  is  made  of  them.  It  is  evident  that  all  computed  lengths  will 
therefore  refer  to  mean  sea  level  without  further  reduction. 

The  stations  forming  a  triangulation  system  are  called  triangu- 
lation stations.  Those  stations  (usually  triangulation  stations) 
at  which  special  work  is  done  are  commonly  given  corresponding 
names,  such  as  base-line  stations,  astronomical  stations,  latitude 
stations,  longitude  stations,  azimuth  stations,  etc. 

An  example  of  a  small  triangulation  system  (United  States 
and  Mexico  Boundary  Survey,  1891-1896)  is  shown  in  Fig.  1, 
page  6,  the  object  being  to  connect  the  "  Boundary  Post "  on 
the  azimuth  line  to  the  westward  with  "  Monument  204  "  on  the 
azimuth  line  to  the  eastward.  The  air-line  distance  between 
these  points  is  about  23  miles.  The  system  is  made  up  of  the 
quadrilateral  West  Base,  Azimuth  Station,  East  Base,  Station 
No.  9;  the  quadrilateral  Pilot  Knob,  Azimuth  Station,  Station 
No.  10,  Station  No.  9;  the  quadrilateral  Pilot  Knob,  Azimuth 
Station,  Station  No.  10,  Monument  204;  and  the  triangle  Pilot 
Knob,  Boundary  Post,  Azimuth  Station.  The  base  line  (West 
Base  to  East  Base)  has  a  length  of  2,205  meters  ( 1. 37 -f  miles), 
and  the  successive  expansions  are  evident  from  the  figure. 

9.  Geometrical  Conditions.  The  triangles  and  combinations 
thereof  which  make  up  a  triangulation  system  form  a  figure  involv- 
ing rigid  geometrical  relations  among  the  various  lines  and  angles. 
The  measured  values  seldom  or  never  exactly  satisfy  these  con- 


GEODETIC  SURVEYING 

114140' 


Boundary  Post 


LUW 

Pilot  Knob 


Azimuth  M  Station. 
West  Case. 


Sta.  No.  9.  Sta.  No/10. 


Triangulation 

in  vicinity  of 

Yuma,  Arizona; 

International  Boundary  Survey 

United  States  and  Mexico, 

1891-1896. 


FIG.  1.— Example  of  a  Triangulation  System. 

From  Report  of  U.  S.  Section  of  International  Boundary  Commission. 


PRINCIPLES  OF  TRIANGULATION  7 

ditions,  and  must  therefore  be  adjusted  until  they  do.  In  the 
nature  of  things  the  true  values  Jof  the  lines  and  angles  can  never 
be  known,  but  the  greater  the  number  of  independent  conditions 
on  which  an  adjustment  is  based  the  greater  the  probability  that 
the  adjusted  values  lie  nearer  to  the  truth  than  the  measured 
values.  It  is  for  this  reason  that  work  of  an  extended  character 
is  arranged  so  that  some  or  all  of  the  measured  values  will  be 
involved  in  more  than  one  triangle,  thus  greatly  increasing  the 
number  of  conditions  which  must  be  satisfied  by  the  adjustment. 

The  simplest  system  of  triangulation  is  that  in  which  the  work 
is  expanded  or  carried  forward  through  a  succession  of  independent 
triangles,  each  of  which  is  separately  adjusted  and  computed; 
and  where  the  work  is  of  moderate  extent  this  is  usually  all  that  is 
necessary.  The  best  triangulation  system,  under  ordinary  circum- 
stances, when  the  survey  is  of  a  more  extended  character,  or 
great  accuracy  is  desired,  is  that  in  which  the  work  is  so  arranged 
as  to  form  a  succession  of  independent  quadrilaterals,  each  of 
which  is  separately  adjusted  and  computed.  (In  work  of  great 
magnitude  the  entire  system  would  be  adjusted  as  a  whole.) 
A  geodetic  quadrilateral  is  the  figure  formed  by  connecting  any 
four  stations  in  every  possible  way,  the  result  being  the  ordinary 
quadrilateral  with  both  its  diagonals  included;  there  is  no  station 
where  the  diagonals  intersect.  The  eight  corner  angles  of  the 
quadrilateral  are  always  measured  independently,  and  then 
adjusted  (as  explained  later)  so  as  to  satisfy  all  the  geometric 
requirements  of  such  a  figure.  Other  arrangements  of  triangles 
are  sometimes  used  for  special  work.  More  complicated  systems 
of  triangles  or  adjustment  are  seldom  necessary  or  desirable, 
except  in  the  very  largest  class  of  work.  Since  triangulation 
systems  are  usually  treated  as  a  succession  of  independent  figures 
it  evidently  makes  no  difference  whether  the  figures  overlap  or 
extend  into  new  territory. 

Every  triangulation  system  is  fundamentally  made  up  of 
triangles,  and  in  order  that  small  errors  of  measurement  shall  not 
produce  large  errors  in  the  computed  values,  it  is  necessary  that 
only  well  shaped  triangles  should  be  permitted.  The  best  shaped 
triangle  is  evidently  equilateral,  while  the  best  shaped  quadri- 
lateral is  a  perfect  square,  and  these  are  the  figures  which  it  is 
desirable  to  approximate  as  far  as  possible.  A  well  shaped 
triangle  is  one  which  contains  no  angle  smaller  than  30°  (involving 


8 


GEODETIC  SURVEYING 


the  requirement  that  no  angle  must  exceed  120°).  In  a  quadri- 
lateral, however,  angles  much  less  than  30°  are  often  necessary 
and  justifiable  in  the  component  triangles. 

10.  Special  Cases.  It  is  often  desirable  and  feasible  (espe- 
cially on  reconnoissance)  to  connect  two  distant  stations  with  a 
narrow  and  approximately  straight  triangulation  system,  as  shown 
diagrammatically  by  the  several  plans  in  Fig.  2.  In  these  diagrams 
the  heavy  dots  represent  the  stations  occupied,  all  the  angles  at 
each  station  being  directly  measured.  The  maximum  length 


II 


III 


FIG.  2. 

of  sight  is  approximately  the  same  in  each  case.  The  stations 
to  be  connected  are  marked  A  and  B.  In  an  actual  survey,  of 
course,  the  location  of  the  stations  could  only  approximate  the 
perfect  regularity  of  the  sketches. 

In  System  I  the  terminal  stations  are  connected  by  a  simple 
chain  of  triangles.  This  plan  is  the  cheapest  and  most  rapid, 
but  also  the  least  accurate. 

System  II  is  given  in  two  forms,  which  are  substantially  alike 
in  cost  and  results,  the  hexagonal  idea  being  the  basis  of  each 
construction.  This  system  not  only  covers  the  largest  area, 
but  greatly  increases  the  accuracy  attainable.  The  large  num- 


PRINCIPLES  OF  TEIANGULATION  9 

ber  of  stations  in  this  system  necessarily  increases  both  the  labor 
and  the  cost.  & 

System  III  is  formed  by  a  continuous  succession  of  quadri- 
laterals, and  is  the  one  to  use  where  the  highest  degree  of  accuracy 
is  desired.  The  area  covered  is  less  than  in  System  I,  but  the  cost 
and  labor  approximate  System  II. 

11.  Classification  of  Triangulation  Systems.  It  has  been 
found  convenient  to  classify  triangulation  systems  (and  the 
triangles  involved)  as  primary,  secondary  and  tertiary,  based  on 
the  magnitude  and  accuracy  of  the  work. 

Primary  triangulation  is  that  which  is  of  the  greatest  magnitude 
and  importance,  sometimes  extending  over  an  entire  continent. 
In  work  of  this  character  the  highest  attainable  degree  of  accuracy 
(1  in  500,000  or  better)  is  sought,  using  long  base  lines,  large  and 
well  shaped  triangles,  the  highest  grade  of  instruments,  and  the 
best  known  methods  of  observation  and  computation.  Primary 
base  lines  may  measure  from  three  to  ten  or  more  miles  in  length, 
with  successive  base  lines  occurring  at  intervals  of  one  hundred 
to  several  hundreds  of  miles  (about  30  to  100  times  the  length 
of  base) ,  depending  on  the  character  of  the  country  traversed  and 
the  instrument  used  in  making  the  measurement.  In  primary 
triangulation  the  sides  of  the  triangles  may  vary  from  20  to  100 
miles  or  more  in  length. 

Secondary  triangulation  covers  work  of  great  importance, 
often  including  many  hundred  miles  of  territory,  but  where  the 
base  lines  and  triangles  are  smaller  than  in  primary  systems,  and 
where  the  same  extreme  refinement  of  instruments  and  methods 
is  not  necessarily  required.  An  accuracy  of  1  in  50,000  is  good 
work.  Base  lines  in  secondary  work  may  measure  from  one  to 
three  miles  in  length,  and  occur  at  intervals  of  about  twenty  to 
fifty  times  the  length  of  base.  The  triangle  sides  may  vary  from 
about  five  to  forty  miles  in  length. 

Tertiary  triangulation  includes  all  those  smaller  systems 
which  are  not  of  sufficient  size  or  importance  to  be  ranked  as 
primary  or  secondary.  The  accuracy  of  such  work  ranges  upwards 
from  about  1  in  5,000.  The  base  lines  measure  from  about  a 
half  to  one  and  a  half  miles  long,  occurring  at  intervals  of  about 
ten  to  twenty -five  times  the  length  of  base.  The  triangle  sides  may 
measure  from  a  fraction  of  a  mile  up  to  about  six  miles  in  length. 

In  an  extended  survey  the  primary  triangulation  furnishes 


10  GEODETIC  SURVEYING 

the  great  main  skeleton  on  which  the  accuracy  of  the  whole  survey 
depends;  the  secondary  systems  (branching  from  the  primary) 
furnish  a  great  many  well  located  intermediate  points;  and  the 
tertiary  systems  (branching  from  the  secondary)  furnish  the 
multitude  of  closely  connected  points  which  serve  as  the  reference 
points  for  the  final  detailed  work  of  the  survey. 

12.  Selection  of  Stations.    This  part  of  the  work  calls  for  the 
greatest  care  and  judgment,  as  it  practically  controls  both  the 
accuracy  and  the  cost  of  the  survey.     Every  effort,  therefore, 
should  be  made  to  secure  the  best  arrangement  of  stations  con- 
sistent with  the  object  of  the  survey,  the  grade  of  work  desired, 
and  the  allowable  .cost.     The  base  line  is  usually  much  smaller 
than  the  principal  lines  of  the  triangulation  system,  and  there- 
fore requires  an  especially  favorable  location,  in  order  that  its 
length    may    be    accurately    determined.     Approximately    level 
or  gently  sloping  ground   (not  over  about  4°)   is  demanded  for 
good  base-line  work.     It  is  also  necessary  that  the  base  line  be 
connected  as  directly  as  possible  with  one  of  the  main  lines  of  the 
system,  using  a  minimum  number  of  well  shaped  triangles.     The 
base-line  stations  and  the  connecting  triangulation  stations  are 
consequently  dependent  on  each  other,  in  order  that  both  objects 
may  be  served.     In  flat  country  the  greatest  freedom  of  choice 
would   probably  lie  with  the  base-line  stations,  while  in  rough 
country   the  triangulation   stations  would   probably   be   largely 
controlled  by  a  necessary  base-line  location. 

The  various  stations  in  a  triangulation  system  must  be  selected 
not  only  with  regard  to  the  territory  to  be  covered  and  the  for- 
mation of  well  shaped  triangles,  but  so  as  to  secure  at  a  minimum 
expense  the  necessary  intervisibility  between  stations  for  the 
angles  to  be  measured.  Clearing  out  lines  of  sight  is  expensive 
in  itself,  and  may  also  result  in  damages  to  private  interests. 
Building  high  stations  in  order  to  see  over  obstructions  is  like- 
wise expensive.  A  judicious  selection  of  stations  may  materially 
reduce  the  cost  of  such  work  without  prejudicing  the  other 
interests  of  the  survey.  It  is  important  that  lines  of  sight  should 
not  pass  over  factories  or  other  sources  of  atmospheric  disturb- 
ance. These  and  similar  points  familiar  to  surveyors  must  all 
receive  the  most  careful  consideration. 

13.  Reconnoissance.    The    preliminary    work    of    examining 
the  country  to  be  surveyed,  selecting  and  marking  the  various 


PEINCIPLES  OF  TRIANGULATION  11 

base-line  and  angle  stations,  determining  the  required  height 
for  tower  stations,  etc., "is  called  reconnaissance .  As  much  infor- 
mation as  possible  is  obtained  from  existing  maps,  such  as  the 
height  and  relative  location  of  probable  station  points  and  desir- 
able arrangement  of  triangles.  The  reconnoissance  party  then 
selects  in  the  field  the  best  location  of  stations  consistent  with  the 
grade  and  object  of  the  survey  and  in  accordance  with  the  prin- 
ciples laid  down  in  the  preceding  article.  The  reconnoissance 
is  often  carried  forward  as  a  survey  itself,  so  that  fairly  good 
values  are  obtained  of  all  the  quantities  which  will  finally  be 
determined  with  greater  accuracy  by  the  main  survey.  When  a 
point  is  thought  to  be  suitable  for  a  station  a  high  signal  is  erected, 
such  as  a  flag  on  a  pole  fastened  on  top  of  a  tree  or  building, 
and  the  surrounding  country  is  scanned  in  all  directions  to  pick 
up  previously  located  signals  and  to  select  favorable  points  for 
advance  stations. 

The  instrumental  outfit  of  the  reconnoissance  party  is  selected 
in  accordance  with  the  character  of  the  information  which  it 
proposes  to  obtain.  In  any  event  it  must  be  provided  with 
convenient  means  for  measuring  angles,  directions,  and  eleva- 
tions. A  minimum  outfit  would  probably  contain  a  sextant  for 
measuring  angles,  a  prismatic  compass  for  measuring  directions, 
an  aneroid  barometer  for  measuring  elevations,  a  good  field  glass, 
and  creepers  for  climbing  poles  and  trees. 

A  common  problem  ior  the  reconnoissance  party  is  to  estab- 
lish the  direction  between  two  stations  which  can  not  be  seen 
from  each  other  until  the  forest  growth  is  cleared  out  along 
the  connecting  line.  Any  kind  of  a  traverse  run  from  one  station 
to  the  other  would  furnish  the  means  for 
computing  this  direction,  but  the  follow- 
ing simple  plan  can  often  be  used : 

Let  AB,  Fig.  3,  be  the  direction  it  is 
desired  to  establish.  Find  two  inter- 
visible  points  C  and  D  from  each  of 
which  both  A  and  B  can  be  seen. 
Measure  each  of  the  two  angles  at  C 
and  D  and  assume  any  value  (one  is 
the  simplest)  for  the  length  CD.  From 

the  triangle  ACD  compute  the  relative  value  of  AD.  Sim- 
ilarly from  BCD  get  the  relative  value  of  BD.  Then  from  the 


12 


GEODETIC  SURVEYING 


triangle  A BD  compute  the  angles  at  A  and  B,  which  will  give 
the  direction  of  A  B  from  either  end  with  reference  to  the  point 
D.  All  computed  lengths  are  necessarily  only  relative  because 
CD  was  assumed,  but  the  computed  angles  are  of  course  correct. 

The  required  intervisibility  of  any  two  stations  must  be  finally 
determined  on  the  ground  by  the  reconnoissance  party,  but  a 
knowledge  of  the  theoretical  considerations  governing  this  ques- 
tion is  of  the  greatest  importance  and  usefulness. 

14.  Curvature  and  Refraction.  Before  discussing  the  inter- 
visibility  of  stations  it  is  necessary  to  consider  the  effect  of  curva- 
ture and  refraction  on  a  line  of  sight.  In  geodetic  work  curvature 


FIG.  4. 

is  understood  to  mean  the  apparent  reduction  of  elevation  of 
an  observed  station,  due  to  the  rotundity  of  the  earth  and 
consequent  falling  away  of  a  level  line  (see  Art.  76)  from  a 
horizontal  line  of  sight.  Refraction  is  understood  to  mean  the 
apparent  increase  of  elevation  of  an  observed  station,  due  to 
the  refraction  of  light  and  consequent  curving  of  the  line  of  sight 
as  it  passes  through  air  of  differing  densities.  The  net  result 
is  an  apparent  loss  of  elevation,  causing  an  angle  of  depression 
in  sighting  between  two  stations  of  equal  altitude.  In  Fig.  4 
the  circle  ADE  represents  a  level  line  through  the  observing 
point  A,  necessarily  following  the  curvature  of  the  earth.  Assum- 


PKINCIPLES  OF  TRIANGULATION 


13 


ing  the  line  of  sight  to  be  trul£  level  or  horizontal  at  the  point 
A,  the  observer  apparently  sees  in  the  straight  line  direction 
AB  (tangent  to  the  circle  at  A),  but  owing  to  the  refraction  of 
light  actually  looks  along  the  curved  line  AC  (also  tangent  at 
A) .  The  observer  therefore  regards  C  as  having  the  same  eleva- 
tion as  A,  whereas  the  point  D  is  the  one  which  really  has  the 
same  elevation  as  A.  There  is  hence  an  apparent  loss  of  eleva- 
tion at  C  equal  to  CD,  as  the  net  result  of  the  loss  BD  due  to 
curvature  and  the  gain  BC  due  to  refraction.  Just  as  C  appears 
to  lie  at  B}  so  any  point  F  appears  to  lie  at  a  corresponding  point 
G.  The  apparent  difference  of  elevation  of  the  points  A  and 
F  is  measured  by  the  line  BG,  the  true  difference  being  DF. 
As  DF  =  BG  +  BD  -  FG,  the  apparent  loss  equals  BD  -  FG, 
which  does  not  ordinarily  differ  much  from  CD. 


FIG.  5. 

So  far  as  the  intervisibility  of  two  stations  is  concerned  it  is 
only  necessary  to  know  the  effect  of  curvature  and  refraction 
with  reference  to  a  straight  line  tangent  to  the  earth  at  mean 
sea  level.  Referring  to  Fig.  5,  BD  represents  the  effect  of 
curvature,  and  BC  the  effect  of  refraction,  as  in  the  previous 
figure.  By  geometry  we  have 

AB2  =  BD  X  BE. 

The   earth   is  so   large   as  compared  with   any  actual   case  in 
practice    that    we    may   substitute  AD    (=distance,    called  K) 


14  GEODETIC  SUKVEYING 

for  AB,  and  DE  ( =  2R)  for  RE,  without  any  practical  error, 
and  write 

Distance2  K2 

RD  =  curvature  =  -r—   — r. —    — ^ r  =  p-yr, 

Aver.  diam.  of  earth      2R ' 

in  which  all  values  are  to  be  taken  in  the  same  units.  (For 
mean  value  of  R  see  Table  X  at  end  of  book.)  As  the  result  of 
proper  investigations  we  may  also  write 

Distance2  K2          K2 

BC  =  refraction  =  ra-r—       — -, — -= -r-  =  m-^-  =  2m— 5 , 

Aver.  rad.  ot  earth         R  2R 

in  which  m  is  a  coefficient  having  a  mean  value  of  .070,  and  K  and 
R  are  the  same  as  before.  (For  additional  values  of  m  see  Art.  85.) 
We  thus  have 

K2 

BD-BC  =CD=  curv.  and  refract.  =  (1  -  2m}— ^ . 

ZrC 

Table  I  (at  end  of  book)  shows  the  effect  of  curvature  and 
refraction,  computed  by  the  above  formula,  for  distances  from 
1  to  66  miles. 

15.  Intervisibility  of  Stations  The  elevation  (or  altitude) 
of  a  station  is  the  elevation  of  the  observing  instrument  above 
mean  sea  level.  This  is  not  to  be  confused  with  the  height  of  a 
station,  which  is  the  elevation  of  the  instrument  above  the  natural 
ground.  In  order  that  two  stations  may  be  visible  from  each 
other  the  line  of  sight  must  clear  all  intermediate  points.  The 
necessary  (or  minimum)  elevation  of  each  station  will  therefore 
be  governed  by  the  following  considerations: 

1.  The  elevation  of  the  other  station.     Obviously  a  line  of  sight 
which  is  required  to  clear  a  given  point  by  a  certain  amount  can 
not  be  lowered  at  one  end  without  being  raised  at  the  other. 

2.  The  profile  of  the  intervening  country.     It  is  evidently  not 
only   the  height  of  an  intermediate  point  but  also  its  location 
between  the  two  stations  that  will  determine  its  influence  on  their 
inter  visibility.     An  elevation  great  enough  to  obstruct  the  line 
of  sight  if  located  near  the  lower  station  might  be  readily  seen 
over  if  located  near  the  higher  station. 

3.  The  distance  between  the  stations.     Owing  to  the  curvature 
of  the  earth  it  is  necessary  in  looking  from  one  point  to  another 
to  see  over  the  intervening  rotundity,  the  extent  of  which  depends 


PEINCIPLES  OF  TRIANGULATION  15 

on  the  distance  between  the  stations.  Since  lines  of  sight  are 
nearly  straight  this  can-  not  be  accomplished  unless  at  least  one 
of  the  stations  has  a  greater  elevation  than  any  intermediate 
point.  Owing  to  the  refraction  of  light  the  line  of  sight  is  not 
really  a  straight  line,  but  in  any  actual  case  is  practically  the  arc 
of  a  circle,  with  the  concavity  downwards,  and  a  radius  about 
seven  times  that  of  the  earth.  This  fact  slightly  lessens  the 
elevation  necessary  to  see  over  the  rotundity,  but  otherwise 
does  not  change  the  conditions  to  be  met.  Thus  in  Fig.  5  the 
points  F  and  C  are  just  barely  intervisible,  though  F  and  C  both 
have  greater  elevations  than  A. 

In  view  of  the  above  facts  it  is  usually  necessary  to  place 
stations  on  the  highest  available  ground,  such  as  ridge  lines, 
summits,  or  mountain  peaks,  increasing  the  height,  if  necessary, 
by  suitably  built  towers. 

The  simplest  question  of  intervisibility  is  illustrated  in  Fig.  5, 
page  13,  where  all  points  between  stations  F  and  C  lie  at  the 
elevation  of  mean  sea  level.  If  the  elevation  of  F  is  given  or  as- 
sumed the  corresponding  distance  HA  to  the  point  of  tangency  is 
taken  out  directly  from  Table  I  (interpolating  if  necessary).  The 
value  CD  corresponding  to  the  remaining  distance  AD  is  then 
taken  out  from  the  same  table,  and  gives  the  minimum  elevation 
of  C  which  will  make  it  visible  from  F.  Thus  if  HD  =  30.0 
miles,  and  elevation  of  F=  97.0  ft.,  we  have  HA  =  13.0  miles, 
and  the  remaining  distance  AD  =  17.0  miles,  calling  for  a  min- 
imum elevation  of  165.8  ft.  for  station  C. 

In  general  the  profile  between  two  stations  is  more  or  less 
irregular,  and  the  question  can  not  be  handled  in  the  above 
simple  manner.  It  is  usually  necessary  to  compute  the  elevations 
of  the  line  of  sight  at  a  number  of  different  points  and  compare 
the  results  with  the  ground  elevation  at  such  points.  The  critical 
points  are  usually  evident  from  an  inspection  of  the  profile. 
Owing  to  the  uncertainties  of  refraction  accurate  methods  of 
computation  are  not  worth  while;  different  methods  of  approx- 
imation give  slightly  different  results,  but  all  sufficiently  near 
the  truth  for  the  desired  purpose. 

The  following  example  will  show  a  satisfactory  method  of  pro- 
cedure in  any  case  that  may  arise  in  practice.  The  line  AEJP, 
Fig.  6,  page  16,  is  the  natural  profile  of  the  ground,  and  it  is  desired 
if  possible  to  establish  stations  at  A  and  P.  The  critical  points 


16 


GEODETIC  SURVEYING 


that  might  obstruct  the  line  of  sight  are  evidently  at  E  and  /. 
Assume  the  following  data  to  be  known: 

Distances  (at  mean  sea  level) .     Elevations  (above  M.  S.  L.). 
BH  =  30.0  miles  A  =  1 140.6  ft.  =  AB 

\Q.l     "  E  =  1322.7  "  =EH 

W.7     "  J  =  1689.0  t:  =  JN 

P=  2098.3  "  =PR 

For  an  imaginary  line  of  sight  BQ,  horizontal  at  B   we  have 
from  Table  I  (by  interpolating): 


Elevation  of 


G  =    516.4  ft.  =GH. 

M  =    922.8  "  =MN.     Hence  PQ  =  617.4.  ft. 
Q  =  1480.9  "  =QR. 


FIG.  6. 


Assuming  the  lines  of  sight  BP,  AP,  and  AO  to  have  the  same 
radius  of  curvature  as  BQ,  we  may  write  approximately 


BH 
BR 


and 


LM      EM       BN 


PQ      BQ      BR  PQ        BQ   "  BR' 

giving,  by  substitution,  FG=  364.6  ft.  and  LM  =  498.3  ft. 

F  =    881.0  ft. 


Hence  we  have  elevation  of 


L  =  1421.1  ft. 


By  the  similar  approximations 

DF  _  FP  _  HR 
AB~BP~BR     and 


KL 


LP_ 
BP 


BR' 


PRINCIPLES  OF  TRIANGULATION  17 

we  find  DF  =  467.0  ft.  and  KL  ^  240.2  ft. 

s\D  =  1348.0ft. 

Hence  we  nave  elevation  of  {  T^       inG1  0  r, 

[/I  =  lobl.o  it. 

Hence  the  line  of  sight  AP  clears  E  by  25.3  ft.,  but  fails  to 
clear  J  by  27.7  ft. 

16.  Height  of  Stations.     Referring  to  the  previous  article, 
suppose  it  is  desired  to  erect  a  tower  OP,  so  that  the  line  of  sight 
OA  shall  clear  the  obstruction  /.     It  was  found  that  the  line 
PA  failed  to  clear  J  by  27.7  ft.,  and  it  is  not  desirable  to  have  a 
line  of  sight  less  than  6  ft.  from  the  ground,  hence  IK  should  be 
about  34  ft.     Using  the  approximation 

°L  =  AP.  =  ^        °*L  =  50-8 
TK~AK~BN        3C6~5u' 

we  find  OP  =43. 1ft. 

Hence  a  suitable  tower  at  P  should  not  be  less  than  43  ft. 
high.  If  it  were  desired  to  build  a  smaller  tower  at  P,  the  instru- 
ment at  A  would  also  have  to  be  elevated,  the  amount  being 
determined  by  a  similar  plan  of  approximation.  It  is  evident 
that  the  least  total  height  of  towers  is  obtained  by  building  a 
single  tower  at  the  station  nearest  to  the  obstruction.  If  the 
obstruction  is  practically  midway  between  the  stations  the  com- 
bined height  of  any  two  corresponding  towers  would  of  course 
come  the  same  as  that  of  a  suitable  single  tower.  If  more  than 
one  obstruction  is  to  be  seen  over,  the  most  economical  arrange- 
ment of  towers  is  readily  found  by  a  few  trial  computations. 

In  heavily  wooded  country  tower  stations  extending  above  the 
tree  tops  are  frequently  more  economical  than  clearing  out  long 
lines  of  sight,  and  their  construction  is  therefore  justified  even 
though  the  intervening  country  would  not  otherwise  demand 
their  use.  In  general  it  is  not  wise  to  have  a  line  of  sight  near 
the  ground  for  any  large  portion  of  its  length,  on  account  of  the 
unsteadiness  of  the  atmosphere  and  the  risk  of  sidewise  refraction. 

17.  Station    Marks.     Any  kind  of  a  survey  requires  the  station 
marks  to  remain  unchanged  at  least  during  the  period  of  the 
survey.     When  work  is  of  sufficient  magnitude  or  importance 
to  justify  geodetic  methods  and  instruments,  permanent  station 
marks  are  usually  desirable.     The  best  plan  seems  to  be  to  place 
the  principal  mark  below  the  ground,  as  least  likely  to  suffer 


18  GEODETIC  SURVEYING 

disturbance  by  frost,  accident,  or  malicious  interference.  Though 
many  plans  have  been  tried,  the  common  underground  mark 
consists  of  a  stone  about  6"X6"X24"  placed  vertically  with 
its  top  about  30"  below  the  surface  of  the  ground,  the  center 
point  being  marked  by  a  small  hole  or  copper  bolt.  The  under- 
ground mark  is  of  course  only  used  in  case  there  is  reason  to 
think  the  surface  mark  has  been  moved.  The  surface  mark 
usually  consists  of  a  similar  stone,  reaching  nearly  down  to  the 
bottom  stone  and  extending  a  few  inches  above  the  surface, 
with  the  station  point  similarly  marked.  Three  witness  stones 
are  commonly  set  near  the  station  (where  least  likely  to  be  dis- 
turbed, ordinarily  200  or  more  feet  from  the  station,  and  forming 
approximately  an  equilateral  triangle),  with  their  azimuths 
and  distances  recorded,  so  that  the  station  might  be  restored 
if  entirely  destroyed.  Stones  about  36"  long  and  projecting 
about  12"  above  the  surface  have  proven  satisfactory.  Other 
means  of  establishing  permanent  stations  will  suggest  themselves 
to  the  surveyor  when  the  surrounding  conditions  are  known. 

18.  Observing  Stations  and  Towers.     In  addition  to  the  station 
mark  a  suitable  support  is  required  to  carry  the  observing  instru- 
ment.    Unless  the  tripod  is  very  heavy  and  stiff  it  will  not  prove 
satisfactory.     In  such  a  case  a  rigid  support  must  be  provided. 
Heavy  posts  well  set  in  the  ground  may  serve  as  the  basis  for 
such  a  construction  for  a  low  height,  bracing  as  may  prove  neces- 
sary for  rigidity.     If  an  observing  platform  is  built  it  must  not 
be  connected  in  any  way  with  the  structure  that  carries  the  instru- 
ment.    A  low  masonry  pier  makes  an  excellent  station.     Under 
15  ft.  in  height  a  tripod  can  be  built  at  the  station  heavy  enough 
to  be  satisfactory  as  an  instrument  support.     For  greater  heights 
a  regular  tower  should  be  built  to  carry  the  instrument,  so  braced 
and  guyed  as  to  be  absolutely  immovable  and  free  from  vibra- 
tion.    The  observer's  platform  must  be  carried  by  an  entirely 
independent  structure  surrounding  the  instrument  tower  with- 
out being  in  any  way  connected  with  it,  or  in  any  way  possible 
to  come  in  contact  with  it.     A  light  awning  on  a  framework 
attached  to  the  observer's  platform  should  shelter  the  instrument 
from  the  sun.     Fig.  7  shows  a  common  form  of  tower  station. 

19.  Station  Signals  or  Targets.     These  terms  (used  more  or 
less  interchangeably)  refer  to  that  object  at  a  station  which  is 
sighted  at  by  observers  at  other  stations.     A  satisfactory  target 


PRINCIPLES  OF  TRIANGULATION 


19 


FIG.  7.— Tower  Station. 

From  Appendix  No.  9,  Report  for  1882,  U.  S.  C.  and  G.  S. 


20  GEODETIC  SURVEYING 

must  be  distinctly  visible  against  any  background  and  of  suit- 
able width  for  accurate  bisection,  and  preferably  free  from  phase. 
When  the  face  of  a  target  is  partially  illuminated  and  partially 
in  shadow,  the  observer  usually  sees  only  the  illuminated  portion 
and  thus  makes  an  erroneous  bisection,  the  apparent  displace- 
ment of  the  center  of  the  target  being  called  phase.  Targets  of 
this  kind  have  been  used  and  rules  for  correction  for  phase  devised, 
but  targets  free  from  phase  are  much  to  be  preferred.  The  target 
may  be  a  permanent  part  of  the  station  (such  as  a  flagpole  carried 
by  an  overhead  construction  so  as  to  clear  the  instrument),  or 
only  brought  into  service  when  the  station  is  not  occupied  (such 
as  flagpoles,  heliotropes  or  night  signals).  In  any  case  a  signal 
must  of  course  be  accurately  centered  over  the  station.  Eccentric 
signals  are  sometimes  used,  involving  a  corresponding  reduction 
of  results,  but  where  the  instrument  and  signal  can  not  occupy 
the  same  position  it  is  more  common  to  regard  the  signal  as  the 
true  station  and  the  instrument  as  eccentric. 

Board  Signals.  Approximately  square  boards,  three  or  more 
feet  wide,  painted  in  black  and  white  vertical  stripes  or  other 
designs,  have  been  tried  as  targets  and  found  usually  unsatis- 
factory, except  for  distances  of  a  few  miles  only.  The  painted 
designs  are  hard  to  see  unless  in  direct  sunlight  and  not  easy  to 
bisect  even  then.  They  present  their  full  width  in  only  one 
direction.  If  two  such  boards  are  placed  at  right  angles  (whether 
as  a  cross  or  one  above  the  other)  so  as  to  give  a  good  apparent 
width  in  any  direction,  the  shadow  of  one  board  on  the  other 
produces  the  very  phase  difficulty  that  board  targets  were  designed 
to  prevent. 

Pole  Signals.  Round  (sometimes  square)  poles,  painted  black 
and  white  in  alternate  lengths,  are  frequently  used  for  signals. 
Against  a  sky  background  they  give  good  results,  but  against 
a  dark  background  they  may  give  the  usual  trouble  from  phase. 
Their  diameter  should  be  about  1J  inches  for  the  first  mile, 
increasing  roughly  as  the  square  root  of  the  distance.  Their  size 
becomes  prohibitory  for  distances  of  over  15  or  20  miles.  The 
equivalent  of  a  pole  signal,  made  out  of  wire  and  canvas  and  free 
from  phase,  was  found  very  satisfactory  on  the  Mississippi  River 
Survey.  The  general  construction  consisted  of  four  vertical 
wires  forming  a  square,  held  in  place  by  wire  rings  (all  con- 
nections soldered),  black  and  white  canvas  being  stretched 


PRINCIPLES  OF  TRIANGULATION 


21 


across  the  diagonal  wires  between^the  successive  rings,  so  as  to 
form  a  vertical  series  of  Black  and  white  planes  at  right  angles 
to  each  other  and  showing  both  colors  in  both  directions.  The 
distance  between  the  rings  was  made  several  times  the  diameter 
of  the  rings,  so  that  any  shadow  or  phase  effect  would  affect  only 
a  very  small  part  of  the  length  of  each  canvas.  In  addition  to 
being  accurately  centered  any  pole  or  equivalent  signal  must  of 
course  be  set  truly  vertical. 

Heliotropes.  When  the  distance  between  stations  exceeds 
about  15  or  20  miles  resort  is  had  to  reflected  sunlight  as  a  signal. 
If  the  reflecting  surface  is  of  proper  size  such  a  signal  is  entirely 
satisfactory  for  any  distance  from  the  smallest  to  the  largest, 
on  account  of  the  certainty  with  which  it  is  seen.  Any  device 


To  Station 


FIG.  8. — Heliotrope. 

by  which  the  rays  of  the  sun  may  be  reflected  in  a  given  direction 
is  called  a  heliotrope,  the  essential  features  being  a  plane  mirror 
and  a  line  of  sight.  A  simple  form  of  such  an  instrument  is 
shown  in  Fig.  8.  An  additional  mirror  (called  the  back  mirror) 
is  also  required,  in  order  to  reflect  the  sunlight  onto  the  main 
mirror  when  it  can  not  be  directly  received.  The  heliotrope  is 
generally  mounted  on  a  tripod,  with  a  horizontal  motion  for 
lining  in  with  the  distant  station,  and  is  centered  over  its  own 
station  with  a  plumb  bob. 

In  more  elaborate  forms  a  telescope  with  universal  motion 
furnishes  the  line  of  sight,  the  mirror  and  vanes  being  mounted 
on  top  of  it. 

In  using  the  instrument  it  is  pointed  towards  the  observing 


22  GEODETIC  SURVEYING 

station  by  means  of  the  sight  vanes  or  telescope,  and  the  mirror 
is  turned  so  as  to  throw  the  shadow  of  the.  near  vane  centrally 
on  the  farther  vane,  an  attendant  moving  the  mirror  slightly 
every  few  minutes  as  required.  The  cone  of  rays  reflected  by 
the  mirror  subtends  an  angle  of  about  32  minutes  (the  angular 
diameter  of  the  sun  as  seen  from  the  earth),  or  about  50  feet 
in  width  per  mile.  The  light  will  therefore  be  seen  at  the  observ- 
ing station  if  the  error  of  pointing  is  less  than  16  minutes  or  about 
25  feet  per  mile.  The  topographical  features  of  the  country 
generally  enable  the  heliotroper  to  locate  a  station  with  this  degree 
of  approximation  without  any  other  aid,  though  it  is  well  to  be 
provided  with  a  good  pair  of  field  glasses  if  the  heliotrope  has  no 
telescope.  The  observing  station  usually  has  a  heliotrope  also, 
so  that  the  two  stations  may  be  in  communication  by  agreed 
signals  or  by  using  the  telegraphic  alphabet  of  dots  and  dashes 
(long  flashes  for  dashes  and  short  ones  for  dots,  swinging  a  hat  or 
other  handy  object  in  front  of  the  mirror  to  obscure  the  light  as 
desired) .  When  each  station  has  a  heliotrope  they  soon  find  each 
other  by  swinging  the  light  around  slowly  until  either  one  catches 
the  other's  light,  when  the  two  heliotropes  are  quickly  and 
accurately  centered  on  each  other. 

The  best  size  of  mirror  to  use  depends  on  the  character  of  the 
observing  instrument,  the  state  of  the  atmosphere,  and  the  dis- 
tance between  stations.  In  order  to  have  a  signal  capable  of 
accurate  bisection  it  must  be  neither  dangerously  indistinct  nor 
dazzlingly  bright.  Between  these  limits  there  is  a  wide  range 
of  light  which  is  satisfactory.  If  the  light  is  too  bright  it  is 
readily  reduced  by  covering  the  mirror  with  a  cardboard  disc 
containing  a  suitable  sized  hole.  A  mirror  whose  diameter  is 
proportioned  at  the  rate  of  0.2  inch  per  mile  of  distance  will 
answer  well  for  average  conditions  of  climate  and  instruments. 
In  the  dry  climate  of  our  western  states  one-half  this  rate  will 
prove  sufficient.  In  the  southern  part  of  California  the  writer 
has  seen  a  six-inch  mirror  for  80  miles  across  the  Yuma  desert  with 
the  naked  eye,  but  this  required  exceptionally  favorable  conditions. 

The  apparent  size  of  the  heliotrope  light  varies  remarkably 
with  the  time  of  day  and  the  condition  of  the  atmosphere,  this 
phenomenon  being  an  actual  measurable  fact  and  not  an  optical 
illusion.  At  sunrise  and  sunset  the  light  appears  as  small  as  a 
star,  almost  covered  by  the  vertical  hair,  and  giving  a  perfect 


PRINCIPLES  OF  TRIANGULATION  23 

I 

pointing.  Anywhere  within  about^wo  hours  of  sunrise  and  sunset 
the  image  is  circular,  clean  cut,  and  readily  bisected,  the  size 
of  the  image  increasing  rapidly  with  the  distance  of  the  sun  above 
the  horizon.  After  the  sun  has  risen  a  couple  of  hours  above  the 
horizon  until  noon  the  image  gradually  be3OTies  more  and  more 
irregular  in  outline  and  gains  in  size  at  an  enormous  rate,  some- 
times filling  25  per  cent  of  the  field  of  view  of  the  telescope  at 
noon.  The  image  then  decreases  in  size  and  becomes  gradually 
more  regular  in  outline,  becoming  fit  to  observe  again  about  two 
hours  before  sunset.  When  the  wind  blows  strongly  the  image 
elongates  like  an  ellipse,  and  appears  to  wave  and  nutter  like 
a  flag.  If  the  attendant  neglects  his  work,  so  that  either  the 
back  mirror  or  main  mirror  is  poorly  pointed,  the  image  loses 
rapidly  in  brilliancy.  On  the  United  States  Boundary  Survey, 
however,  it  was  found  by  the  most  careful  micrometric  experi- 
ments that  the  center  of  the  apparent  image  always  corresponded 
with  the  true  center  of  station. 

Only  one  objection  has  been  urged  against  the  heliotrope, 
namely,  that  it  can  only  be  used  when  the  sun  is  shining,  while 
angles  are  best  measured  on  cloudy  days.  Nevertheless,  the 
heliotrope  furnishes  the  best  solution  for  long  distance  signals 
in  the  daytime,  and  good  results  can  be  obtained  by  making  the 
measurements  close  to  sunrise  and  sunset.  For  the  best  class 
of  work  the  afternoon  period  is  much  fhe  best,  as  great  risk  of 
sidewise  (lateral)  refraction  always  endangers  the  work  of  the 
morning  period. 

Night  signals.  A  great  deal  of  geodetic  work  has  been  done 
at  night,  using  an  artificial  light  as  a  signal,  aided  by  a  lens  or 
parabolic  reflector.  Up  to  about  forty  miles  a  kerosene  light  with 
an  Argand  burner  is  entirely  satisfactory.  For  any  practicable 
distance  the  acetylene  gas  lamp  is  found  to  meet  every  require- 
ment. Other  kinds  of  lights  have  been  successfully  used,  but 
those  above  given  have  the  advantage  that  only  unskilled  labor 
is  required  to  operate  them,  such  as  can  operate  heliotropes  in 
the  daytime.  Up  to  midnight  fully  as  good  work  can  be  done 
as  in  the  daytime,  but  the  remainder  of  the  night  does  not  pro- 
vide favorable  atmospheric  conditions  for  close  work.  The  chief 
advantage  of  night  work  is,  of  course,  the  fact  that  it  practically 
doubles  the  number  of  hours  per  day  available  for  good  work. 


CHAPTER   II 
BASE-LINE  MEASUREMENT 

20.  General  Scheme.  The  accurate  measurement  of  base 
lines  required  for  geodetic  work  may  be  accomplished  with  rigid 
base-bars  placed  successively  end  to  end,  or  with  flexible  wires 
or  tapes  stretched  successively  from  point  to  point.  Base-bars 
were  formerly  used  exclusively  for  the  highest  grade  of  work, 
but  tape  or  wire  measurements  are  rapidly  growing  in  favor. 
The  invar  tape  (a  special  kind  of  steel  tape)  is  now  being  used  ex- 
clusively by  the  United  States  Coast  and  Geodetic  Survey  for  its 
base  line  measurements.  The  convenience  of  the  steel  tape  is 
apparent,  and  the  ease  and  rapidity  with  which  it  can  be  used 
are  strong  points  in  its  favor. 

No  form  of  measuring  apparatus  maintains  a  constant  length 
at  all  temperatures,  nor  is  it  often  possible  to  measure  along  a 
mathematically  straight  line.  Base  lines  can  seldom  be  located  at 
sea  level.  The  actual  length  of  a  bar  or  tape  under  standard 
conditions  (called  its  absolute  length)  is  seldom  found  to  be 
exactly  the  same  as  its  designated  length.  Tapes  and  wires  are 
elastic,  and  their  length  varies  with  the  tension  (pull)  under 
which  they  are  used.  The  weight  of  tapes  or  wires  (when  unsup- 
ported) causes  them  to  sag  and  thus  draw  the  ends  closer  together. 
In  base-bar  work  corrections  may  hence  be  required  for  absolute 
length,  temperature,  horizontal  and  vertical  alignment,  and  reduc- 
tion to  mean  sea  level.  With  tape  or  wire  measurements  correc- 
tions may  be  required  for  absolute  length,  temperature,  pull, 
sag,  horizontal  and  vertical  alignment,  and  reduction  to  mean  sea 
level.  These  corrections  will  be  considered  in  turn  after  describ- 
ing the  types  and  use  of  bars  and  tapes. 

21.  Base-bars  and  Their  Use.  The  fundamental  idea  of  a 
base-bar  is  a  rigid  measuring  unit,  such  as  a  metallic  rod.  The 
general  scheme  of  measuring  a  base  requires  the  use  of  two 
such  bars.  The  first  bar  is  placed  in  approximate  position, 

24 


BASE-LINE  MEASUREMENT  25 

supported  at  the  quarter  points  t^y  two  tripods  or  trestles,  care- 
fully aligned  both  horizontally  and  vertically,  and  moved  longi- 
tudinally forward  or  backward  until  its  rear  end  is  vertically 
over  one  end  of  the  base  line.  The  second  bar,  similarly  supported 
and  aligned,  is  then  drawn  longitudinally  backward  until  its  rear 
end  is  just  in  contact  with  the  forward  end  of  the  first  bar.  The 
first  bar  and  its  supports  are  then  carried  forward,  alignment  and 
contact  made  as  before,  and  the  measurement  so  continued  to 
the  end  of  the  base.  In  the  simple  form  outlined  above  the 
method  would  not  produce  results  of  sufficient  accuracy  for 
geodetic  work,  but  with  the  perfected  methods  and  appa- 
ratus in  actual  use  measurements  of  extreme  precision  may  be 
made. 

Several  features  are  more  or  less  common  to  all  types  of  base- 
bar.  The  actual  measuring  unit  is  generally  made  of  metal  and 
protected  by  an  outer  casing  of  wood  or  metal.  Mercurial 
thermometers  are  located  inside  the  casing  for  temperature 
measurements.  Means  are  provided  for  aligning  the  bars  hori- 
zontally, usually  a  telescope  suitably  mounted  at  the  forward 
end  of  the  bar.  Vertical  alignment  is  provided  for,  usually 
by  a  graduated  sector  carrying  a  level  bubble,  mounted  on  the 
side  ot  the  bar  near  its  central  point,  so  that  the  bar  may  be  made 
truly  horizontal  or  its  inclination  determined.  A  slow  motion 
is  provided  for  making  the  contact  wTith  the  previous  bar;  the 
slow  motion  is  produced  by  turning  a  milled  head  at  the  rear  of 
the  bar,  which  moves  the  measuring  unit  only,  the  casing  remain- 
ing stationary  in  its  approximate  position  on  the  tripods  on 
account  of  the  friction  due  to  the  weight  of  the  bar.  The  rod 
(or  tube)  constituting  the  measuring  unit  terminates  at  its  for- 
ward end  with  a  small  vertical  abutting  plane;  the  rear  end 
of  the  rod  carries  a  sliding  sleeve  pressed  outward  by  a  light 
spring  and  ending  in  a  small  straight  knife  edge  for  making  the 
contact  with  the  abutting  plane  of  the  previous  bar;  the  length 
of  the  bar  is  the  distance  between  the  knife  edge  and  the  abutting 
plane  of  its  measuring  unit  when  the  sliding  sleeve  is  in  its  proper 
place,  indicated  by  a  mark  on  the  sleeve  coinciding  with  a  mark 
on  the  rod;  the  forward  bar  is  therefore  brought  into  proper 
position  without  disturbing  the  rear  bar,  the  only  pressure  on 
the  rear  bar  being  that  due  to  the  light  spring  controlling  the 
contact  sleeve  while  the  forward  measuring  unit  is  slowly  drawn 


26  GEODETIC  SURVEYING 

backward  until  the  coincidence  of  the  indicating  lines  shows  that 
the  bar  is  in  its  proper  place. 

One  of  the  earlier  forms  of  bar  used  by  the  U.  S.  Coast  and 
Geodetic  Survey  is  described  in  Appendix  No.  17,  Report  for  1880, 
and  called  a  perfected  form  of  a  contact-slide  base  apparatus. 
This  bar  was  an  improvement  on  similar  bars  in  previous  use, 
and  besides  the  features  enumerated  above  contained  a  new 
device  for  determining  its  own  temperature.  The  actual  measuring 
unit  was  a  steel  rod  8  mm.  in  diameter.  A  zinc  tube  9.5  mm. 
in  diameter  was  placed  on  each  side  of  the  steel  rod  (not  quite 
reaching  either  end) .  The  rear  end  of  one  zinc  tube  wTas  soldered 
to  the  rear  end  of  the  steel  rod,  and  the  forward  end  of  the  other 
zinc  tube  was  soldered  to  the  forward  end  of  the  steel  rod.  By 
suitable  scales  on  the  steel  rod  and  the  free  ends  of  the  zinc  tubes 
the  apparatus  was  thus  converted  into  a  metallic  thermometer 


FIG.  9. — Thermometric  Base-bar. 

(zinc  having  a  coefficient  of  expansion  about  2J  times  that  of 
steel),  so  that  the  temperature  of  the  bar  became  very  accurately 
measured.  In  Fig.  9  the  arrangement  is  shown  in  outline, 
the  light  line  indicating  steel  and  the  heavy  lines  zinc.  This  bar 
was  4  meters  long. 

In  Appendix  No.  7,  Report  for  1882,  a  compensating  bar  is 
described.  This  bar  is  made  of  a  central  zinc  rod  and  two  side 
steel  rods,  as  shown  in  Fig.  10.  The  ends  of  this  bar  remain 


FIG.  10.— Compensating  Base-bar. 

nearly  the  same  distance  apart  at  all  temperatures.  The  com- 
pensation is  not  absolutely  perfect,  however,  and  the  scales  at 
each  end  indicate  the  temperature  so  that  the  final  small  correc- 
tion may  be  made  for  this  cause.  This  bar  was  5  meters  long. 
In  Appendix  No.  11,  Report  for  1897,  the  Eimbeck  duplex 
base-bar  is  described,  this  bar  having  almost  entirely  superseded 
those  previously  discussed.  This  bar  is  a  bi-metallic  contact- 


BASE-LINE  MEASUREMENT  27 

slide  apparatus  consisting  of  two  measuring  units  of  precisely 
similar  construction,  one  x>f  steel  and  one  of  brass,  each  5  meters 
in  length,  and  weighs  complete  118  pounds.  The  measuring 
units  are  made  of  tubing  f  inch  in  diameter,  each  having  a 
thickness  of  wall  corresponding  to  the  conductivity  and  specific 
heat  of  the  material  of  which  it  is  made,  so  that  under  changing 
conditions  each  tube  shall  keep  the  same  temperature  as  the 
other  one,  which  is  an  essential  requirement.  The  two  measuring 
tubes  are  carried  in  a  brass  protecting  casing,  which  turns  on 
its  longitudinal  axis  in  an  outer  brass  protecting  casing  which 
remains  stationary.  The  inner  casing  is  rotated  180°  from  time 
to  time  to  equalize  temperature  distribution.  This  bar  is  illus- 
trated in  Figs.  11  and  12.  The  two  measuring  units  are  entirely 
disconnected,  and  contact  is  always  made  brass  to  brass  and  steel 
to  steel,  so  that  two  independent  measures  of  the  base  are 
obtained,  one  by  the  brass  unit  and  one  by  the  steel  unit.  The 
difference  in  the  length  of  these  two  measurements  furnishes 
the  key  to  the  average  temperature  of  the  bars  during  the 
measuring,  so  that  the  correction  for  temperature  can  be  very 
closely  determined.  Since  the  coefficient  of  expansion  for  brass 
is  about  H  times  that  for  steel,  the  two  measuring  units  are 
seldom  of  the  same  length,  and  the  shorter  one  continually 
gains  on  the  longer  one.  To  overcome  this  difficulty  the  meas- 
uring units  are  provided  with  vernier  scales,  and  the  brass 
bar  is  occasionally  shifted  a  small  amount  which  is  read  from 
the  scales  and  recorded  for  an  evident  purpose.  The  duplex  bar 
is  superior  to  the  bars  previously  described  both  in  speed 
and  accuracy.  A  speed  of  forty  bars  per  hour  is  readily  main- 
tained. 

The  tripods  used  to  support  base-bars  must  be  absolutely 
rigid.  Special  heads  are  provided  so  that  both  quick  and  slow 
motion  are  available  for  raising  the  bar  support.  The  rear  tripod 
usually  has  a  knife-edge  support  and  the  front  one  a  roller  sup- 
port. By  easing  the  weight  on  the  edge  support  the  bar  may 
be  readily  moved  on  the  roller  support  and  quickly  brought  into 
proper  position. 

Satisfactory  work  is  accomplished  with  base-bars  at  all  hours 
of  the  day.  In  order  to  protect  the  bars  from  the  extreme  heat 
of  the  sun,  however,  a  portable  awning  is  often  placed  over  them, 
which  is  dragged  steadily  forward  as  the  work  advances. 


28 


GEODETIC  SURVEYING 


PQ  02 

X  b 

<o  ** 


BASE-LINE  MEASUREMENT 


29 


30  GEODETIC  SURVEYING 

22.  Steel  Tapes  and  Their  Use.  Steel  tapes  for  base-line  work 
do  not  differ  materially  from  ordinary  tapes  except  in  length. 
Surveyors  generally  use  tapes  50  or  100  feet  long,  and  with 
proper  precautions  a  high  grade  of  work  can  be  done.  Better 
or  quicker  work,  however,  can  probably  be  done  with  longer 
tapes,  such  tapes  usually  being  also  somewhat  smaller  in  cross- 
section.  Experience  shows  that  tapes  300  to  500  feet  in  length 
and  with  about  0.0025  square  inch  cross-section  are  entirely 
satisfactory. 

It  is  seldom  desirable  to  use  the  tape  directly  on  the  ground, 
on  account  of  the  uneven  surface  and  the  uncertainties  of  fric- 
tion. The  usual  way  is  to  support  the  tape  at  a  number  of  equi- 
distant points  (20  to  100  feet),  letting  it  hang  suspended  between 
these  points  and  computing  the  corresponding  correction  for 
sag.  In  order  to  avoid  any  friction  the  supports  are  usually 
wire  loops  swinging  from  nails  driven  in  carefully  aligned  stakes. 
Unless  the  points  of  support  are  on  an  even  and  determined 
grade  it  is  necessary  to  measure  the  elevation  of  each  such  point, 
in  order  to  make  the  necessary  reduction  for  vertical  alignment, 
that  is,  reduction  to  the  horizontal.  The  points  of  support 
must  have  such  elevations  that  the  pull  on  the  tape  will  not 
lift  it  free  of  any  of  the  supports.  No  change  of  horizontal 
alignment  is  allowable  within  a  single  tape  length.  It  is  evident 
that  good  work  can  not  be  done  with  a  suspended  tape  if  an 
appreciable  wind  is  blowing. 

The  pull  on  the  tape  must  be  exerted  through  the  medium 
of  a  spring  balance  or  other  device  attached  to  the  forward  end. 
The  pull  adopted  may  be  from  12  to  20  pounds,  depending  on  the 
weight  of  the  tape  and  the  distance  between  supports,  so  as  to 
prevent  excessive  sagging  and  to  hold  the  tape  in  line.  For  an 
accuracy  of  1  in  50,000  the  pull  may  be  made  with  a  good  spring 
balance,  properly  steadied  by  connection  with  a  good  stake. 
For  extreme  accuracy  the  pull  must  be  known  within  a  question 
of  ounces,  and  special  stretching  devices  attached  to  firmly  driven 
stakes  are  required.  The  desired  amount  of  pull  can  be  very 
accurately  made  through  the  simple  device  of  a  weight  acting 
through  a  right-angled  lever  turning  on  a  knife-edge  fulcrum; 
the  device  must  be  so  mounted  that  the  lever  arms  can  be  brought 
into  a  truly  vertical  and  horizontal  position  when  the  strain  is 
on  the  tape. 


BASE-LINE  MEASUREMENT  31 

The  length  of  a  steel  tape  is  materially  modified  by  a  moderate 
change  of  temperature^  so  that  the  greatest  care  is  required  in 
making  the  corresponding  correction.  It  is  found  in  practice 
that  a  high  grade  of  work  can  not  be  done  in  direct  sunlight, 
owing  to  the  difficulty  of  ascertaining  the  temperature  of  the  tape, 
a  mercurial  thermometer  held  near  the  tape  or  in  contact  with  it 
failing  to  give  the  true  value  by  many  degrees.  An  accuracy 
of  1  in  50,000  requires  the  mean  temperature  of  the  tape  to  be 
known  within  a  degree,  and  an  accuracy  of  1  in  500,000  to  within 
one-fifth  of  a  degree.  The  highest  grade  of  work  can  therefore 
be  done  only  on  densely  cloudy  days  or  at  night. 

In  the  common  method  of  using  steel  tapes  the  tape  is  stretched 
(suspended)  between  two  tripods  (or  posts  driven  or  braced  until 
immovable),  the  rear  one  being  carried  forward  in  turn  for  each 
new  tape  length.  Intermediate  supports  are  provided  as  previ- 
ously described,  if  necessary.  The  rear  end  of  the  tape  is  con- 
nected with  a  straining  stake  a  few  feet  back  of  the  rear  tripod; 
the  front  end  is  connected  with  the  spring  balance  or  other  device 
for  giving  the  desired  pull,  the  strain  at  this  end  also  being 
resisted  by  a  suitable  stake  or  stakes  beyond  the  forward  tripod ; 
in  this  way  no  strain  is  allowed  to  come  on  either  tripod.  A 
small  strip  of  zinc  is  secured  to  the  top  of  each  tripod,  and  each 
tripod  is  set  with  sufficient  care  so  that  the  end  mark  on  the 
tape  will  come  somewhere  on  the  zinc  strip,  the  exact  point  being 
marked  by  making  a  fine  scratch  on  the  zinc  with  any  suitable 
instrument.  In  regard  to  temperature  measurements  tapes  100 
feet  or  less  in  length  ought  to  have  two  thermometers  tied  to 
them,  one  at  each  quarter  point;  longer  tapes,  up  to  about  300 
feet,  ought  to  be  equipped  with  three  thermometers,  one  at  the 
center,  and  one  about  one-sixth  the  length  from  each  end. 

Professor  Edward  Jaderin  of  Stockholm  has  obtained  the  very 
best  results  in  a  method  slightly  differing  from  the  above.  Profes- 
sor Jaderin  prefers  a  tape  25  meters  long,  5  centimeters  each 
side  of  the  25-meter  mark  being  graduated  to  millimeters  and 
read  by  estimation  to  the  nearest  tenth  of  a  millimeter.  Each 
tripod  carries  a  single  fixed  graduation,  and  the  distance  between 
the  marks  on  two  successive  tripods  must  not  vary  more  than  5 
centimeters  either  way  from  25  meters.  By  means  of  the  end 
scale  on  the  tape  the  exact  distance  from  tripod  to  tripod  is 
determined  and  the  whole  base  found  by  the  sum  of  the  results. 


32  GEODETIC  SURVEYING 

The  best  work  can  only  be  done  on  densely  cloudy  days  or  at 
night. 

23.  Invar  Tapes.     By  alloying  steel  with  about  35  per  cent 
of  nickel  a  material  is  produced  possessing  an  exceedingly  small 
coefficient  of  expansion,  this  discovery  being  due  to  C.  E.  Guil- 
laume   (of  the  International  Bureau  of  Weights  and  Measures, 
near  Paris).     For  this  reason  the  name  "  invar  "  (from  "inva- 
riable ")  has  been  applied  to  this  material.     Tapes  made  of  invar 
have  proven  extremely  satisfactory  for  the  accurate  measure- 
ment of  base  lines,  errors  in  determining  the  temperature  of  the 
tape  being  of  so  much  less  importance  than  with  steel  tapes,  which 
makes  it  possible  to  do  first  class  work  at  all  hours  of  the  day. 

The  coefficient  of  expansion  of  invar  is  about  1:28  that  of 
steel,  or  about  0.00000022  per  degree  Fahrenheit.  The  modulus 
of  elasticity  is  about  8: 10  that  of  steel,  or  about  23,000,000  pounds 
per  square  inch.  The  tensile  strength  is  about  100,000  pounds 
per  square  inch,  or  about  half  that  of  the  ordinary  steel  tape,  but 
amply  sufficient  for  the  purpose.  The  yield  point  is  about  70 
per  cent  of  the  tensile  strength. 

In  1905  the  Coast  Survey  purchased  six  invar  tapes  from 
J.  H.  Agar  Baugh,  London,  Eng.,  for  the  purpose  of  subjecting 
them  to  the  actual  test  of  field  work  and  comparing  them  with 
steel  tapes  under  similar  conditions.  (See  Appendix  No.  4, 
1907.)  These  tapes  averaged  about  0".02XO".25  in  cross- 
section,  about  53  meters  in  length,  looked  more  like  nickel  than 
steel,  and  were  full  of  innumerable  small  kinks  which,  however, 
did  not  cause  any  inaccuracy  in  actual  service.  They  were  very 
soft  and  easily  bent,  being  much  less  elastic  than  steel,  and  requir- 
ing reels  16  inches  in  diameter  to  prevent  permanent  bending. 
Steady  loads  up  to  60  pounds  caused  no  permanent  set.  While 
rusting  more  slowly  than  steel  tapes  they  were  found  to  need  oiling 
and  care. 

The  experience  of  the  Coast  Survey  with  invar  tapes  indicates 
that  they  possess  no  properties  derogatory  to  their  use  for  base- 
line work,  and  that  under  similar  conditions  both  better  and 
cheaper  work  can  be  done  than  with  steel  tapes.  They  are  used 
in  all  respects  like  steel  tapes,  using  special  care  to  avoid  injury 
from  bending. 

24.  Measurements  with  Steel   and   Brass  Wires.      Professor 
Edward  Jaderin  of  Stockholm  has  found  it  possible  to  do  excellent 


BASE-LINE  MEASUREMENT  33 

base-line  work  throughout  the  entire  day  by  using  steel  and  brass 
wires  instead  of  steel' tapes.  (See  U.  S.  C.  and  G.  S.  Appendix 
No.  5,  Report  for  1893.)  The  object  of  using  the  metal  in  wire 
form  instead  of  tape  form  is  to  minimize  the  effect  of  the  wind, 
since  the  circular  cross-section  (for  the  same  area)  exposes  much 
less  surface  to  the  action  of  the  wind  than  the  flat  surface  of 
the  tape  form.  The  method  used  is  the  same  as  described  in  the 
last  paragraph  of  Art.  22,  except  that  two  values  are  obtained 
for  the  distance  between  each  pair  of  tripods,  one  with  the  steel 
and  one  with  the  brass  wire.  Two  measurements  of  the  whole 
base  line  are  thus  obtained,  and  from  their  difference  the  average 
temperature  of  the  wires  is  deduced  and  hence  the  corresponding 
correction.  The  assumption  is  made  that  the  wires  are  always  of 
equal  temperature,  both  being  given  the  same  surface  (nickel 
plate,  for  example),  the  same  cross-section,  and  the  same  hand- 
ling. The  principle  is  identical  with  that  of  the  Eimbeck 
duplex  base-bar  described  in  Art.  21. 

25.  Standardizing  Bars  and  Tapes.  The  nominal  length  of  a 
bar  or  tape  is  its  ordinary  designated  length,  as,  for  example,  a 
fifty-foot  tape  or  a  five-meter  bar.  The  actual  length  seldom 
equals  the  nominal  length,  but  varies  with  changing  conditions. 
The  absolute  length  is  the  actual  length  under  specified  conditions. 
If  the  absolute  length  is  known,  the  laws  governing  the  change  of 
length  with  changing  conditions,  and  the  particular  conditions 
at  the  time  of  measuring,  then  the  actual  length  of  the  measuring 
unit  becomes  known,  and  consequently  the  actual  length  of  the 
line  measured.  By  standardizing  a  bar  or  tape  is  meant  deter- 
mining its  absolute  length.  Such  an  expression  as  the  "  tem- 
perature at  which  a  bar  or  tape  is  standard  "means  the  tempera- 
ture at  which  the  actual  and  designated  lengths  agree. 

The  absolute  length  of  a  bar  or  tape  may  be  determined  in  a 
number  of  ways,  but  the  essential  principle  in  each  case  is  the 
same,  namely,  the  comparing  of  the  unknown  length  with  some 
known  standard  length  at  an  accurately  known  temperature. 
If  the  comparison  is  made  in-doors,  the  room  must  be  one  (such 
as  in  the  basement  of  a  building)  where  the  temperature  remains 
practically  constant  for  long  periods,  so  that  the  temperature  of 
the  measuring  units  will  be  the  same  as  that  of  the  surrounding 
air.  If  the  comparison  is  made  in  the  open  air  the  work  must  be 
done  on  a  densely  cloudy  day  or  at  night,  for  the  same  reason. 


34  GEODETIC  SURVEYING 

Tapes  are  standardized  unsupported,  or  supported  horizontally 
throughout  their  length,  at  any  convenient  pull  and  temperature, 
the  Coast  Survey  reducing  the  results  by  computation  to  a  stand- 
ard pull  of  10  pounds  and  temperature  of  62°  F.  The  absolute 
length  of  a  tape  may  be  found  by  measuring  it  with  a  shorter 
unit  (such  as  a  standard  yard  or  meter  bar) ;  by  comparing  it 
with  a  similar  tape  whose  absolute  length  is  known ;  by  comparing 
it  with  fixed  points  whose  distance  apart  is  accurately  known; 
or  by  measuring  with  it  a  base  line  whose  length  is  already  accur- 
ately known.  For  a  nominal  fee  the  Coast  Survey  at  Washington 
will  determine  the  absolute  length  of  any  tape  up  to  100  feet  in 
length. 

Any  device  or  apparatus  which  permits  a  measuring  unit  to 
be  compared  with  a  standard  length  is  called  a  comparator.  It 
is  quite  common  at  the  commencement  of  a  survey  to  fix  two 
points  at  a  permanent  and  well  determined  distance  apart,  and 
compare  all  tapes  used  with  these  points  from  time  to  time;  the 
standard  or  reference  distance  thus  established  would  be  called 
a  comparator.  In  the  laboratory  the  comparator  may  be  a  very 
elaborate  piece  of  apparatus  with  micrometer  microscopes,  by 
which  the  most  accurate  comparisons  may  be  made,  or  with 
which  a  measuring  unit  may  be  most  accurately  measured  by  a 
shorter  standard. 

Base-bars  are  probably  most  readily  and  accurately  standard- 
ized by  measuring  a  base  line  of  known  length  with  them.  The 
actual  length  of  the  bar  thus  becomes  known,  by  computation, 
for  the  temperature  at  which  the  measurement  was  made;  and 
by  means  of  its  coefficient  of  expansion  its  length  becomes  known 
at  any  temperature. 

Measuring  the  same  base  with  the  same  bar  or  tape,  at  widely 
different  temperatures,  furnishes  a  good  means  of  determining 
the  coefficient  of  expansion  if  it  is  not  otherwise  known.  With 
the  compensating  bar  the  coefficient  of  the  residual  expansion 
(since  the  compensation  is  never  perfect)  may  be  thus  obtained. 

If  a  base  line  of  known  length  is  measured  with  a  duplex 
base-bar  at  a  certain  average  temperature,  the  average  actual 
length  of  each  component  bar  (steel  and  brass)  becomes  known 
for  that  temperature,  and  the  difference  in  these  average  lengths 
indicates  that  particular  temperature  and  that  particular  length  of 
each  bar.  The  absolute  length  of  each  component  is  thus  known 


BASE-LINE  MEASUREMENT  35 

for  that  particular  temperature .  If  the  same  thing  is  done  at  a 
widely  different  temperature  the  same  information  is  obtained 
at  the  new  temperature.  Since  the  average  length  of  each  com- 
ponent is  obtained  at  the  two  different  temperatures  the  coefficient 
of  expansion  of  each  component  becomes  known.  Since  the  differ- 
ence in  the  lengths  of  the  components  is  known  at  two  widely 
separated  temperatures,  and  since  this  difference  changes  uniformly 
from  the  lower  to  the  higher  temperature,  the  temperature  corre- 
sponding to  any  particular  difference  in  the  length  of  the  bars  also 
becomes  known.  In  measuring  an  unknown  base  w.th  a  duplex 
bar  (provisionally  using  the  absolute  length  of  each  component 
at  the  standard  temperature  on  which  the  coefficient  of  expansion 
is  based)  the  total  difference  by  the  two  component  bars  becomes 
known,  hence  the  average  difference  per  bar  length,  hence  the 
average  temperature,  hence  by  combination  with  the  coefficient 
of  expansion  the  actual  length  of  each  component  at  the  time 
of  measurement,  hence  the  actual  length  of  the  base  line.  The 
result  must,  of  course,  be  the  same  whether  finally  deduced  from 
the  steel  or  from  the  brass  component,  thus  furnishing  a  good 
check  on  the  computations.  When  base  lines  are  measured 
with  steel  and  brass  wires  these  wires  are  standardized  and  used 
in  the  same  manner  as  the  duplex  base-bar. 

A  base  line  of  known  length,  to  be  used  for  standardizing 
bars  or  tapes,  may  be  one  that  is  measured  with  apparatus  already 
standardized,  or  one  measured  with  a  base-bar  packed  in  melting 
ice  so  as  to  ensure  a  constant  and  known  temperature. 

26.  Corrections  Required  in  Base-line  Work.  As  explained 
in  Art.  20,  if  a  base  line  is  measured  with  base -bars  corrections 
may  be  required  for  absolute  length,  temperature,  horizontal  and 
vertical  alignment,  and  reduction  to  mean  sea  level.  If  the  base 
line  is  measured  with  supported  tapes  or  wires  an  additional 
correction  may  be  required  for  pull.  If  unsupported  tapes  or 
wires  are  used  additional  corrections  may  be  required  for  both 
pull  and  sag.  With  a  simple  or  a  compensating  base-bar,  there- 
fore, it  is  necessary  to  know  its  absolute  length  and  coefficient 
of  expansion  before  it  can  be  used  for  base-line  work.  With  a 
duplex  base-bar  (and  correspondingly  with  double  wire  measure- 
ments) it  is  necessary  to  know  the  absolute  length  and  coefficient 
of  expansion  of  each  of  the  component  units.  With  tapes  and 
wires  it  is  necessary  to  know  the  absolute  length,  coefficient  of 


36  GEODETIC  SURVEYING 

expansion,  modulus  of  elasticity,  area  of  cross-section,  and  weight. 
Except  in  work  of  great  accuracy  average  values  may  be  assumed 
for  the  weight,  coefficient  of  expansion,  and  modulus  of  elasticity 
for  the  material  of  which  the  wire  or  tape  is  made. 

The  above  corrections  are  relatively  so  small  that  they  may  be 
computed  individually  from  the  uncorrected  length  of  base  line, 
and  their  algebraic  sum  taken  as  the  total  correction  required.  A 
plus  correction  means  that  the  uncorrected  length  is  to  be  increased 
to  obtain  the  true  length,  and  a  minus  correction  the  reverse. 

27.  Correction  for  Absolute  Length.  The  absolute  length  of 
a  measuring  unit  is  generally  stated  as  its  designated  length  plus 
or  minus  a  correction.  The  total  correction  will  have  the  same 
sign,  and  be  equal  to  the  given  correction  multiplied  by  the  num- 
ber of  tape  or  bar  lengths  in  the  base  (including  fractional  lengths 
expressed  in  decimals);  or  what  amounts  to  the  same  thing, 
multiply  the  given  correction  by  the  length  of  the  base  and  divide 
by  the  length  of  the  measuring  unit. 

If  Ca=  correction  for  absolute  length; 
c=  correction  to  measuring  unit; 
I  =uncorrected  length  of  measuring  unit; 
L  =uncorrected  length  of  base; 
then 


C**y 
a r. 


Lc 

I 


In  duplex  measurements  the  absolute  lengths  are  used  directly 

in  the  computations  in  order  to  determine  the  average  temperature. 

The  quantities  L  and  I  must  be  expressed  in  the  same  unit 

(feet  or  meters,  for  instance),  and  Ca  will  be  in  the  same  unit  as  c 

(which  need  not  be  the  same  as  used  for  L  and  I) . 

28.  Correction  for  Temperature.  In  measuring  a  base  line 
the  temperature  usually  varies  more  or  less  during  the  progress 
of  the  work,  but  it  is  found  entirely  satisfactory  to  apply  a 
correction  due  to  their  average  temperature  to  the  sum  of  all 
the  even  bar  or  tape  lengths,  and  add  a  final  correction  for  any 
fractional  lengths  and  corresponding  temperatures. 
If  Ct  =  correction  for  temperature  ; 

a  =  coefficient  of  expansion ; 
Tm  —  mean  temperature  for  length  L; 
T8  =  temperature  of  standardization : 
L  =  length  to  be  corrected ; 


BASE-LINE  MEASUREMENT  37 

then  practically,  since  the  measuring  unit  changes  length  uni- 
formly with  the  temperature, 

Ct  ^a(T.m-T8)L. 

Ct  will  be  in  the  same  unit  as  L  and  must  be  applied  with  its 
algebraic  sign. 

The  coefficient  of  expansion  for  steel  wires  and  tapes  may  vary 
from  0.0000055  to  0.0000070  per  degree  F.,  and  if  its  value  is 
not  known  for  any  particular  case  may  be  assumed  as  0.0000063 
(Coast  Survey  value).  For  the  most  accurate  work  the  coeffi- 
cient of  expansion  for  the  particular  tape  or  wire  ought  to  be 
carefully  determined,  either  in  the  laboratory  or  by  measuring  a 
known  base  at  widely  different  temperatures. 

The  coefficient  of  expansion  for  brass  wires  was  found  by 
Professor  Jaderin  to  average  0.0000096  per  degree  F. 

The  coefficient  of  expansion  of  invar  may  be  0.00000022  per 
degree  F.,  or  less. 

In  the  case  of  duplex  measurements  the  average  temperature 
and  corresponding  corrections  may  be  deduced  as  follows : 

Let  Ls  =  provisional  length  of  base,  using  absolute  length  of 
steel    component    at   the    standard    temperature 
(usually  32°  F.  or  0°  C.)  to  which  coefficient  of  ex- 
pansion refers; 
Lb  =  same  for  brass; 
As  =  coefficient  of  expansion  of  steel; 
Ab  =  same  for  brass; 

T  =  average    number    of    degrees  temperature    above 
standard ; 

then  the  true  length  of  base  in  terms  of  steel  component 

=  L8  +  L8A8T, 

and  in  terms  of  brass  component 

=  Lb  +  LbAbT. 

Equating  and  reducing,  we  have 

Ls-Lb 


T  = 


LbAb-L8A8> 


38  GEODETIC  SURVEYING 

and  correction  for  steel-component  measurement 


r        TAT        a8  8 

\-sta  —  ^a^a1    =  ~J  —  7  -  7  —  ~A  —  , 

LbAb—L8A9 
or  practically 

Ct8  =  correction  to  measurement  by  steel  component 


and  similarly 

Ctb  =  correction  to  measurement  by  brass  component 


These  corrections  will  be  in  the  same  unit  as  L8  and  Lb  and  are  to 
be  used  with  their  algebraic  signs. 

29.  Correction  for  Pull.  This  correction  only  occurs  with 
tapes  and  wires;  if  the  pull  used  is  not  the  same  as  that  to  which 
the  absolute  length  is  referred  a  corresponding  correction  must 
be  made. 

Let  CP  =  correction  for  pull; 

Pm  =  pull  while  measuring  base  line ; 
P a  =  pull  corresponding  to  absolute  length; 
S  =  area  of  cross-section  of  tape; 
E  =  modulus  of  elasticity  of  tape; 
L  =  uncorrected  length  of  line; 

then  practically 


SE 

If  E  is  taken  in  pounds  per  square  inch,  then  Pm  and  Pa  must 
be  in  pounds,  L  in  inches,  and  S  in  squares  inches,  whence  CP 
will  be  in  inches,  and  is  to  be  applied  with  its  algebraic  sign. 

If  the  cross-section  is  unknown  it  may  readily  be  found  by 
weighing  the  tape  or  wire  (without  the  box  or  reel),  and  finding 
its  volume  by  comparison  with  the  specific  weight  of  the  same 
material.  The  cross-section  then  equals  the  volume  divided  by 
the  length.  The  weight  of  a  cubic  foot  may  be  assumed  as  490 
pounds  for  steel  tapes,  500  pounds  for  steel  wires,  520  pounds 
for  brass  wires,  and  510  pounds  for  invar  tapes. 


BASE-LINE  MEASUKEMENT  39 

If  the  modulus  of  elasticity  is  unknown  it  may  be  found  as 
follows :    Support  the  -tape  horizontally  throughout  its  length, 
and  apply  two  widely  different  pulls,  noting  how  much  the  tape 
changes  in  length  due  to  the  change  in  the  amount  of  pull. 
Let  P8  =  smaller  pull; 
PI  =  larger  pull; 
I  =  length  of  tape; 

lc  =  change  in  length  caused  by  change  in  pull ; 
S  =  cross-section  of  tape; 
E  =  modulus  of  elasticity ; 
then 


E  = 


SI. 


If  PI  and  P8  are  taken  in  pounds,  I  and  lc  in  inches,  and  S  in 
square  inches,  then  E  will  be  in  pounds  per  square  inch. 

Except  for  the  most  accurate  work  E  may  be  assumed  as 
follows : 

for  steel,  E  =  28,000,000  Ibs.  per  sq.  in. 
for  brass,  E  =  14,000,000 
for  invar,  E  =  23,000,000 

30.  Correction  for  Sag.  This  correction  only  occurs  in  the  case 
of  unsupported  tapes  and  wires.  In  any  actual  case  in  practice 
the  catenary  curve  thus  formed  will  not  differ  sensibly  in  length 
from  a  parabola.  The  correction  required  is  the  difference  in 
length  between  the  curve  and  its  chord. 

Let  C8  =  correction  for  sag  for  one  tape  length  ; 

c  =  correction   for   sag   for   the   interval   between   one 

pair  of  supports; 
I  =  length  of  tape; 

d  =  horizontal  distance  between  supports  (for  which  the 
uncorrected   distance   given   by  the  tape  is  used 
in  practice  without  sensible  error)  ; 
v  =  the  amount  of  sag; 
P  =  the  pull; 
w  =  weight  of  a  unit  length  of  tape. 

The  difference  in  length  between  the  arc  and  chord  "of  a  very  flat 
parabola  (such  as  occurs  in  tape  measurements)  is  found  by  the 


40  GEODETIC  SURVEYING 


calculus  to  be  very  nearly  -—  ,  but  the  formula  is  never  used  in 

3d 

this  form  since  it  is  inconvenient  and  unnecessary  to  measure 
v  in  actual  work.  Passing  a  vertical  section  midway  between 
supports,  and  taking  moments  around  one  support,  we  have 


_  wd      d  _  wd2 
=        X      =    ~~' 


8v2  _  d(wd)2  _  d(wd)2 

3d  ~"  ~  ~  ~ 


from  which 


whence 


and  if  there  are  n  intervals  per  tape 

_  nd(wd)2  _    _  l(wdf 
~      ~24P2~  24P2  ' 

The  correction  to  the  whole  base  line  is  found  by  multiplying 
the  correction  per  tape  length  by  the  number  of  whole  tape 
lengths,  and  adding  thereto  the  corrections  for  any  fractional 
tape  lengths  (which  must  be  computed  separately). 

If  w  is  taken  as  pounds  per  inch,  then  P  must  be  taken  in 
pounds  and  d  and  I  in  inches,  whence  C8  will  be  in  inches. 

The  normal  tension  of  a  tape  is  such  a  tension  as  will  cause 
the  effects  of  pull  and  sag  to  neutralize  each  other,  so  that  no 
correction  need  be  made  for  these  effects.  Since  the  effects  of 
pull  and  sag  are  opposite  in  character  (pull  increasing  and  sag 
decreasing  distance  between  ends  of  tape)  such  a  value  can  always 
be  found  by  equating  the  formulas  (for  a  tape  length)  for  sag 
and  for  pull,  and  solving  for  Pn  or  pull  to  be  used  during  measure- 
ment of  line. 

31.  Correction  for  Horizontal  Alignment.  Ordinarily  base 
lines  are  made  straight  horizontally,  but  sometimes  slight  devi- 
ations have  to  be  introduced,  forming  what  is  called  a  broken 
base.  Fig.  13  shows  a  common  case  of  a  broken  base,  a,  6,  and 
0  being  measured,  and  c  found  by  computation,  some  unavoidable 


BASE-LINE  MEASUREMENT  41 

condition  preventing  the  direct  measurement  of  c.      From  trig- 
onometry we  have 

a2  +  b2  +  2ab  cos  0  =  c2, 

so  that  c  can  always  be  found.     If,  however,  0  is  very  smaU 
(say  not  over  3°)  we  may  proceed  as  follows : 

Let  Cbb  =  correction  for  broken  base;  then 

Cbb  =   -  [(a  +  6)  -  c]; 
but 

a2  +  b2  +  2ab  cos  0  =  c2; 

a2  +  b2  -  c2  =   -  2ab  cos  6. 


Adding  2ab  to  both  members 

a2  +  2ab  +  b2  -  c2  =  2ab  -  2ab  cos  0; 

(a  +  b)2  -  c2  =  2ab  (1  -  cos  0). 
Substituting  (1  -  cos  0)  =  2  sin2  J0, 

[(a  +  6)  -  c]  X  [(a  +  6)  +  c]  =  4ab  sin2 
Hence 

~  4a6  sin2  ^0 


(a  +  6)  +  c ' 

If  0  is  very  small  (which  is  practically  always  the  case)  Cbb  will 
be  very  small,  and  we  may  substitute 

sin  ^0  =  %9  sin  V     and     (a  +  6)  +  c  =  2 (a  +  6), 
whence 

a&02        sin2!' 
X 


a  +  6          2 

in  which  0  must  be  expressed  in  minutes,  and  Cbb  will  be  in  the 
same  unit  as  a  and  b. 

'  =  0.00000004231. 


42  GEODETIC  SURVEYING 

32.  Correction  for  Vertical  Alignment.  When  measurements 
are  taken  with  wires  or  tapes  the  elevations  of  the  different 
points  of  support  will  usually  be  different,  though  frequently 
a  number  of  successive  points  may  be  made  to  fall  on  the  same 
grade. 

Let  Zi,  Z2,  etc.,  be  the  successive  lengths  of  uniform  grades; 
hi,  Ji2,  etc.,  be  the  differences  of  elevation  between  the 

successive  ends  of  these  grades; 
ci,  C2,  etc.,  be  the  numerical  corrections  for  the  single 

grades; 

Cg  =  total  correction  for  grade; 
then  for  any  one  grade 

c  =  I  -  \/l2  -  h2, 


c  -  I  =  -      Z2  -  h2, 

&  -  2lc  +  I2  =  I2  -  h2, 

c2  -  2lc  =  -  h2, 

2lc  -c2  =  h2, 

h2 
~  21  -c' 

but  since  c  is  very  small  in  comparison  with  I  we  may  write  with 
sufficient  precision 


whence 


If  the  grade  lengths  are  all  equal,  as,  for  instance,  when  h 
is  taken  at  every  tape  length, 


C,"    -  -..  n  . 

Fractional  tape  lengths  must  be  reduced  separately. 

When  base-bars  are  used  the  angles  of  inclination  are  measured, 
and  the  correction  is  the  same  for  the  same  angle  whether  the 
angle  is  one  of  elevation  or  depression. 


BASE-LINE  MEASUREMENT 


43 


Let  Cg=  grade  correction  Jpr  one  bar  length; 
/  =  length  of  bar; 

0  —  angle  of  inclination  from  the  horizontal; 
then 

Cg  =   -  I  (1  -  cos  6)  =   -  21  sin2  $0. 

If  0  is  less  than  6°  we  may  write  without  material  error 


sin  \Q  =  £0  sin  I', 


whence 


or 


Cg  =  -  0.00000004231  6% 


with  the  understanding  that  0  is  to  be  expressed  in  minutes, 
and  Cg  will  be  in  the  same  unit  as  I.  The  grade  correction  for 
the  entire  line  will  be  the  sum  of  the  individual  corrections  for 
the  several  bar  lengths. 

33.  Reduction  to  Mean  Sea  Level.  In  geodetic  work  all 
horizontal  distances  are  referred  to  mean  sea  level,  that  is,  the 
stations  are  all  supposed  to  be 
projected  radially  (more  strictly, 
normally)  on  to  a  mean-sea-level 
surface,  and  all  distances  are 
reckoned  on  this  surface.  All  the 
angles  of  a  triangulation  system 
are  measured  as  horizontal  angles, 
and  are  not  practically  affected  by 
the  different  elevations  which  the 
various  stations  may  have.  If  the 
lines  which  are  actually  measured 
(bases  and  check  bases)  are  re- 
duced to  mean  sea  level,  all  com- 
puted lines  will  correspond  to  this 
level  without  further  reduction. 
It  is  necessary,  therefore,  to  con- 
nect the  ends  of  base  lines  with 
the  nearest  bench  marks  whose 

elevations  are  known  with  reference  to  mean  sea  level.  (See 
Art.  77  for  determination  of  mean  sea  level.) 


FIG.  14. 


44  GEODETIC  SURVEYING 

Let  Cmsl  =  reduction  to  mean  sea  level; 
r  =  mean  radius  of  earth; 
a  =  average  elevation  of  base  line; 
B  =  length  of  base  as  measured; 
b  =  length  of  base  at  mean  sea  level; 

then,  from  Fig.  14,  page  43, 

r  +  a      r 


Br 


or  since  a  is  always  very  small  as  compared  with  r,  we  may  write 


in  which  a  and  r  must  be  in  the  same  unit,  and  in  which  Cm8i 
will  be  in  the  same  unit  as  B  (need  not  be  in  the  same  unit  as 
for  a  and  r)  . 

r  (in  meters)  =  6,367,465  log.  =  6.8039665. 

r  (in  feet)        =20,890,592  log.  =  7.3199507. 

34.  Computing  Gaps  in  Base  Lines.  Sometimes  an  obstacle 
occurs  which  prevents  the  direct  measurement  of  a  portion  of 
a  straight  base  line,  as,  for  instance,  between  B  and  C  in  Fig.  15. 
In  such  a  case  if  two  auxiliary  points  A  and  D  (on  the  base) 
are  taken,  x  can  be  computed  if  the  distances  a  and  b  and  the 
angles  a,  /?,  and  6  are  measured.  Draw  BE  and  CF  perpendicular 
to  AO,  and  CG  and  BH  perpendicular  to  DO.  Then 

BE       BA  BO  sin  a  a 

=  -^        or 


CF       CA  CO  sin  («+/?)        x  +  a' 

whence 


Also 


BO 

a  sin  (a  +  /?) 

BH 

CO 
BD 

(x  +  a)  sin  a 
BO  sin  (/?  +  6) 

x  +  b 

CG 

CD 

CO  sin  0 

b     ' 

BASE-LINE  MEASUREMENT  45 

i 

whence  ^ 

*  BO       (x  +  b)smd 
CO       b  sin  (p  +  0)m 

Comparing  (1)  and  (2) 

a  sin  (a+p)  =  (x +b)  sin  0. 

(x  +  a)  sin  a        b  sin  (fi+6)' 
or 

(x  +  a)  (x  +  b)  =  —   - —       p;  sin  p j 

sin  a  sm  6 

which  gives 


/a&  sin  (a  +  ft) 

X   —    ~r  \  I  . 

\  sm  a  s 


sin  (P  +  0)       (a  -  b\2      a  +  b 
s'md  \     2         '        2     ' 


It  is  evident  that  good  results  can  not  be  obtained  unless  the 
points  A,  D,  and  0  are  selected  so  as  to  make  a  well  shaped 
figure. 

35.  Accuracy  of  Base-line  Measurements.  The  accuracy 
possible  in  the  determination  of  the  length  of  a  base  line  depends 
on  the  precision  with  which  the  various  constants  of  the  meas- 
uring apparatus  have  been  obtained  and  the  precision  with  which 
the  field  work  is  done.  The  instrumental  constants  can  be 
determined  with  a  degree  of  precision  commensurate  with  the 
highest  grade  of  field  work.  The  precision  attainable  in  the  field 
is  judged  by  making  repeated  measurements  of  the  same  base 
with  the  same  apparatus  and  comparing  the  results.  From  the 
discrepancies  in  these  measurements  the  probable  error  (Chapter 
XIII)  of  the  average  (arithmetic  mean)  of  the  determinations 


46 


GEODETIC  SURVEYING 


is  found  and  compared  with  the  total  length  of  the  line  as  a 
measure  of  the  precision  attained.  This  measure  of  precision 
is  called  the  uncertainty. 

An  exact  comparison  of  the  merits  of  different  base-line 
apparatus  is  manifestly  impossible,  but  under  similar  conditions 
the  following  results  have  been  obtained  : 

Uncertainty  of  Mean  Length  of  Base.  Steel  tapes  in  cloudy 
weather  or  at  night,  1  in  1,000,000  or  better.  Invar  tapes  at  all 
hours,  1  in  1,000,000  or  better.  Steel  and  brass  wires  at  all 
hours,  1  in  1,000,000  or  better.  Ordinary  base-bars,  1  in  2,000,000 
or  better.  Duplex  base-bars,  1  in  5,000,000  or  better. 

The  probable  error  of  a  base  line  is  obtained  as  follows  : 

Let  ra  =  probable  error  of  mean  length; 

MI,  MZ,  etc.  =  value  of  each  determination; 
z  =  mean  length  of  line; 

Ml  -  z  }  -       ./."•. 

,,  }•  etc.,  =  residuals; 

=  sum  of  squares  of  residuals; 
n  =  number  of  measurements 


then 


±  0.6745 


/     Z& 

\n(n  -  1)' 


Example.    Five  measurements  of  a  base  line  were  made: 


Observed  Values. 

Arithmetic  Mean. 

V. 

t>2. 

6871.26ft. 
6871.31  " 
6871.27  " 
6871.30  " 
6871.28  " 

5)1.42 

6871.284ft. 
6871.284  " 
6871.284  " 
6871.284  " 
6871.284  " 

-  0.024 
+  0.026 
-  0.014 
+  0.016 
-  0.004 

0.000 

0.000576 
0.000676 
0.000196 
0.000256 
0.000016 

0.001720 

.284 

The  algebraic  sum  of  the  residuals  is  zero,  as  it  always  should  be.     Then 
for  ra,  the  probable  error  of  the  mean  length,  we  have 


r,  =  ±  0.6745  .  ±  0.0093  ft.; 

5(5  —  1) 

and  for  Ua,  the  uncertainty  of  the  mean  length,  we  have 

_0.0093_        _1_ 

a  "  6871.284  "  738848' 


CHAPTER   III 
MEASUREMENT  OF  ANGLES 

36.  General    Conditions.     Assuming    that    the    stations    and 
signals  have  been  arranged  to  the  best  advantage,  as  described 
in  Chapter  I,  the  finest  grade  of  instruments  and  favorable  atmos- 
pheric conditions  are  required  for  the  highest  grade  of  work.    The 
U.  S.  Coast  and  Geodetic  Survey  does  satisfactory  work  at  all 
hours,  but  it  is  not  easy  to  do  good  work  in  the  middle  of  the 
day.     From  dawn  to  sunrise  (and  within  about  an  hour  after 
sunrise  if  heliotropes  are  used),  and  from  about  four  o'clock  in 
the   afternoon   until   dark,   represent   the  hours   most   desirable 
for  the  highest  grade  of  work;    even  the  early  morning  period 
frequently   proves   unsatisfactory.     In   densely   cloudy   weather 
work  may  be  carried  on  all  day.     If  night  signals  are  used  (see 
Art.  19),  good  work  can  be  done  up  till  about  midnight.     Accu- 
rate results  can  not  be  expected  if  the  instrument  is  exposed 
to  the  direct  rays  of  the  sun  immediately  before  or  during  the 
measurement  of  an  angle.     The  effect  of  the  sun's  rays  is  to 
cause  heat  radiation,  producing  an  apparent  unsteadiness  of  all 
objects  seen  through  the  telescope,  due  to  the  irregular  refraction 
caused   by  the   currents   of   air   of   different   temperatures;     an 
uncertain  amount  of  sidewise  refraction,  even  if  the  unsteadiness 
is  not  sufficient  to  prevent  a  good  bisection  of  the  signal;    a 
disturbance  of  the  adjustments  of  the  instrument  and  bubbles, 
and  an  actual  twisting  of  the  instrument  on  a  vertical  axis,  both 
caused  by  unequal  expansion  and  contraction;    and  a  twisting 
of  the  station  itself  on  a  vertical  axis,  if  it  have  any  particular 
height  (the  twisting  being  generally  toward  the  sun's  movement, 
and  amounting  to  as  much  as  a  second  of  arc  per  minute  on  a 
75-foot  tower). 

37.  Instruments  for  Angular  Measurements.     Two  types  of 
instrument  are  in  use  for  fine  angle  work,  the  Repeating  Instru- 
ment, and  the  Direction  Instrument,  the  latter  being  considered 

47 


48  GEODETIC  SURVEYING 

the  best  in  the  hands  of  well-trained  observers.  If  either  instru- 
ment is  provided  with  a  vertical  arc  or  circle  it  is  called  an 
Altazimuth  Instrument.  The  term  Theodolite  is  frequently  applied 
to  any  large  instrument  of  high  grade,  though  more  correctly 
limited  to  istruments  in  which  the  telescope  can  not  be  reversed 
without  being  lifted  out  of  its  supports  (on  account  of  the  low- 
ness  of  the  standards).  When  an  instrument  has  to  be  reversed 
in  this  manner  the  telescope  must  be  turned  end  for  end  without 
reversing  the  pivots  in  the  wyes.  The  illustrations  are  all  of 
high  grade  instruments,  Fig.  16  being  a  repeating  instrument, 
Fig.  17  a  direction  instrument,  and  Fig.  18  an  altazimuth  instru- 
ment (in  this  case  also  a  repeating  instrument).  In  general, 
geodetic  instruments  are  larger  than  surveyors'  instruments, 
though  experience  has  shown  that  horizontal  circles  greater  than 
10  or  12  inches  in  diameter  offer  no  further  advantage  in  the 
accuracy  of  the  work  that  can  be  done  with  them.  Such  instru- 
ments are  made  of  the  best  available  material  and  with  the  greatest 
care,  the  utmost  care  being  taken  with  the  graduations  and 
the  making  and  fitting  of  the  centers.  Lifting  rings  are  often 
provided  to  avoid  strain  in  handling.  The  instruments  are 
supported  on  three  leveling  screws  (instead  of  four  as  ordinarily 
found  on  surveyors'  transits),  and  in  addition  a  delicate  striding 
level  is  provided  for  direct  application  to  the  horizontal  axis 
of  the  telescope.  All  the  levels  are  more  delicate  than  on  a 
common  transit,  the  plate  levels  running  from  about  10  to  20 
seconds  per  division,  and  the  striding  level  from  1  to  5  seconds  per 
division.  Repeating  instruments  are  usually  read  by  verniers, 
an  8-inch  instrument  reading  to  10  seconds  and  a  10-  or  12-inch 
instrument  even  down  to  5  seconds,  attached  reading  glasses 
of  high  power  taking  the  place  of  the  ordinary  vernier  glass. 
Direction  instruments  generally  read  to  single  seconds,  as  described 
in  detail  later  on.  The  leveling  screws  (which  support  the 
instruments)  are  pointed  at  the  lower  ends  and  rest  in  V-shaped 
grooves,  so  that  they  are  not  constrained  in  any  way.  If  tri- 
pods are  used  the  grooves  are  usually  cut  in  round  foot  plates 
(about  1^  inches  in  diameter)  properly  placed  on  the  tripod 
head  by  the  maker.  Extra  foot  plates  are  often  provided  which 
can  be  screwed  to  piers  or  station  heads  as  desired.  A  trivet 
is  a  device  often  used  for  the  same  purpose,  consisting  of  a  frame 
containing  three  equally-spaced  radial  V-shaped  grooves  cut  in 


MEASUREMENT  OF  ANGLES 


49 


- 


50 


GEODETIC  SURVEYING 


FIG.  17. — Direction  Instrument. 

From  a  photograph  loaned  by  the  U.  S.  C.  and  G   3. 


MEASUREMENT  OF  ANGLES 


51 


FIG.  18. — Altazimuth  Instrument. 
From  a  photograph  loaned  by  the  U.  S.  C.  and  G.  S. 


52  GEODETIC  SURVEYING 

suitable  arms.  A  three-screw  instrument  is  leveled  by  setting 
a  bubble  parallel  to  a  pair  of  leveling  screws  and  bringing  it 
to  the  center  by  turning  that  pair  of  screws  equally  in  opposite 
directions;  the  crosswise  bubble  is  then  leveled  by  using  only 
the  single  screw  that  is  left. 

38.  The  Repeating  Instrument  and  its  Use.  Besides  the 
features  common  to  all  first-class  instruments,  as  described  in 
the  previous  article,  the  repeating  instrument  must  contain  the 
special  feature  of  a  double  vertical  axis  (as  is  always  the  case 
in  the  surveyor's  transit),  thus  permitting  angles  to  be  measured 
by  the  method  of  repetition.  The  fundamental  idea  of  measuring 
an  angle  by  repetition  is  to  measure  the  angle  a  number  of  times 
without  resetting  the  plates  to  zero  between  the  successive 
measurements,  and  dividing  the  accumulated  result  by  the 
number  of  repetitions.  It  was  at  first  thought  that  any  desired 
degree  of  accuracy  could  be  obtained  by  this  method  by  simply 
increasing  the  number  of  repetitions,  but  it  is  now  known  that 
increasing  the  number  of  repetitions  beyond  a  certain  limit  does 
not  improve  the  result,  on  account  of  systematic  errors  introduced 
by  the  instrument  itself,  chiefly  due  to  the  clamping  attach- 
ments. The  method  is  nevertheless  very  meritorious,  and  excel- 
lent work  can  be  done.  The  object  of  the  repetition  is  twofold: 
First,  the  errors  in  the  pointings  tend  to  compensate  each  other, 
and  the  remaining  error  is  largely  reduced  by  the  division; 
Second,  the  accumulated  reading  is  theoretically  correct  to  the 
least  count  of  the  vernier,  and  the  division  by  the  number  of 
repetitions  tends  to  make  the  reduced  value  as  close  as  if  the 
least  count  were  just  that  much  finer.  There  are  two  ways  of 
measuring  an  angle  by  the  method  of  repetition,  each  designed 
to  eliminate  as  far  as  possible  the  various  instrumental  errors, 
but  based  on  somewhat  different  arguments. 

39.  First  Method  with  Repeating  Instrument.  The  common, 
but  not  the  best,  method  consists  in  repeating  the  angle  any 
desired  number  of  times,  measuring  from  the  left-hand  to  the 
right-hand  station,  with  telescope  direct,  and  dividing  by  the 
number  of  repetitions  to  obtain  one  value  of  the  angle;  then 
measuring  the  same  angle  in  the  reverse  direction  (right-hand 
to  left-hand  station),  using  the  same  number  of  repetitions,  but 
with  telescope  reversed,  and  dividing  as  before  to  obtain  a  second 
value  of  the  angle;  the  average  of  the  two  determinations  is  then 


MEASUREMENT  OF  ANGLES  53 

taken  as  the  value  of  the  angle  (as  given  by  that  set,  and  of 
course  as  many  sets  as-desired  may  be  averaged  together).  The 
number  of  repetitions  in  each  set  is  commonly  so  taken  as  to 
make  each  of  the  accumulated  readings  approximately  equal 
to  one  or  more  times  360°,  in  order  to  eliminate  errors  of  gradu- 
ation. If  this  plan  would  require  an  unreasonable  number  of 
repetitions,  a  number  of  smaller  sets  may  be  taken  from  sym- 
metrical points  around  the  graduated  limb,  and  the  results 
averaged.  Thus  four  independent  sets  might  be  taken,  the  start- 
ing point  for  vernier  A  for  each  set  being  respectively  0°,  45°, 
90°,  and  135°.  The  reversal  of  the  telescope  is  designed  to  elimi- 
nate errors  caused  by  imperfect  adjustment  of  the  collimation 
and  the  horizontal  axis  of  the  telescope.  Measuring  in  opposite 
directions  between  stations  is  designed  to  eliminate  errors  caused 
by  the  clamping  apparatus.  The  reading  of  the  instrument  at 
any  time  is  understood  to  be  the  mean  of  the  readings  of  the 
two  verniers,  as  the  eccentricity  of  the  verniers  and  of  the  centers 
is  thus  eliminated.  The  argument  advanced  in  favor  of  this 
method  is  that  reversing  all  the  processes  for  the  second  half 
of  a  set  ought  to  reverse  the  signs  of  the  various  errors,  so  that 
theoretically  they  ought  to  largely  vanish  from  the  mean  value. 
As  this  method  is  not  recommended  it  is  not  given  in  any  further 
detail. 

40.  Second  Method  with  Repeating  Instrument.  In  this 
method,  considered  the  best,  the  instrument  is  always  revolved 
about  its  vertical  axis  in  the  same  direction 
(almost  universally  clockwise),  no  matter  which 
clamp  is  loosened  nor  how  great  the  angle 
through  which  it  must  be  turned  to  point  to 
the  desired  station.  The  fundamental  scheme 
of  this  method  is  to  measure  (see  Fig.  19)  the 
desired  angle  from  A  to  B  (called  the  interior 
angle) ,  and  also  to  measure  the  other  angle  (called 
the  exterior  angle)  from  the  B  the  rest  of  the  way 
around  to  A,  measuring  this  remaining  angle  being 
called  closing  the  horizon.  The  interior  angle  A  to  FIG.  19. 

B  is  repeated  as  many  times  as  desired  with  the 
telescope  direct   (often  called  normal)  and  an  equal  number  of 
times  with  the  telescope  reversed,  and  the  accumulated  reading 
divided  by  the  total  number  of  repetitions  for  the  provisional 


54  GEODETIC  SURVEYING 

value  of  this  angle.  The  exterior  angle  B  to  A  is  measured  in 
exactly  the  same  way  with  the  same  number  of  repetitions, 
etc.  The  values  thus  obtained  for  the  interior  and  exterior 
angles  are  added  together,  and  if  the  result  is  not  exactly  360° 
the  discrepancy  is  equally  divided  between  the  two  angles. 
The  entire  operation  makes  one  set.  The  argument  in  favor 
of  this  method  is  that  since  the  exterior  angle  is  measured  in 
identically  the  same  way  as  the  interior  angle  it  ought  to  be 
subject  to  exactly  the  same  error;  adding  the  two  angles  together, 
therefore,  should  double  the  error;  and  the  value  of  this  double 
error  be  made  apparent  by  the  failure  of  the  sum  to  equal  360°. 
The  assumption  is  evidently  made  that  the  errors  which  it  is 
sought  to  eliminate  by  this  method  are  independent  of  the  size 
of  the  angle,  and  this  is  generally  believed  to  be  true.  In  practice 
the  verniers  are  not  reset  to  zero  after  completing  the  measure- 
ment of  the  interior  angle,  but  become  the  starting  point  for  the 
measurement  of  the  exterior  angle  just  as  they  stand;  the 
instrument  is  thus  made  to  automatically  add  the  interior  and 
exterior  angles  on  its  own  graduations,  and  the  verniers  should 
therefore  read  zero  (360°)  at  the  completion  of  the  set  if  no  errors 
were  involved.  It  is  more  common  for  the  combined  angles  to  run 
under  than  over  360°,  about  10"  per  repetition  not  being  an 
unusual  amount.  It  is  found  by  experience  with  this  method  that 
six  repetitions  (3  direct  and  3  reversed)  of  the  interior  angle,  and 
the  same  for  the  exterior  angle,  make  a  very  satisfactory  set; 
and  the  average  of  two  such  sets  (if  in  close  agreement)  gives 
a  very  good  determination  of  the  desired  angle.  The  plates  are 
not  reset  to  zero  between  the  two  sets,  but  left  undisturbed  as 
a  starting  point  for  the  second  set,  so  that  the  vernier  readings 
become  slightly  different  each  time  and  the  mind  is  free  from 
bias.  The  complete  program  for  a  double  set  would  be  as  follows : 

PROGRAM 
FIRST  SET. 

1.  Level  up,  set  vernier  d  to  zero,  read  vernier  B. 
Set  telescope  direct  and 

2.  Undamped  below,  turn  clockwise  and  set  on  left  station. 

3.  "  above,  "  "          right       " 

4.  Unclamp  below,  and  read  vernier  A. 


MEASUREMENT  OF  ANGLES  55 

Leaving  verniers  unchanged,        & 

5.  Undamped  below,  turn  clockwise  and  set  on  left  station. 

6.  "           above,             "                     "  right 

7.  "          below,             "                     "  left 

8.  "           above,             "                      "  right 

Reverse  telescope  and 

9.  Undamped  below,  turn  clockwise  and  set  on  left  station. 

10.  "           above,             "                      "  right       " 

11.  "           below,              "                      "  left 

12.  "           above,             "                      "  right 

13.  "•          below,             "                     "  left 

14.  "           above,              "                      "  right 

15.  Unclamp  below  and  read  both  verniers. 

Leaving  telescope  reversed  and  verniers  unchanged, 

16.  Undamped  below,  turn  clockwise  and  set  on  right  station. 

17.  "           above,             "                      "  left          " 

18.  "           below,              "                      "  right       " 

19.  "           above,              "                      "  left 

20.  "           below,             "                     "  right       " 

21.  "           above              "                     "  left 

Set  telescope  direct  and 

22.  Undamped  below,  turn  clockwise  and  set  on  right  station. 

23.  "           above,             "                     "  left 

24.  "           below,             "                     "  right       " 

25.  "           above,             "                     "  left 

26.  "          below,             "                     "  right       " 

27.  "           above,             "                     "  left 

28.  Unclamp  below  and  read  both  verniers. 

SECOND  SET. 

1.     Leaving  verniers  unchanged    from    previous  set,    relevel 
with  lower  motion  undamped. 

Set  telescope  direct  and 

2.  Undamped  below,  turn  clockwise  and  set  on  left  station. 

3.  "           above,             "                     "  right       " 

etc.  etc. 


56  GEODETIC  SURVEYING 

40a.  Reducing  the  Notes.  The  following  points  are  taken 
advantage  of  to  save  labor  in  reducing  the  notes: 

First.  In  finding  the  average  value  of  the  six  repetitions 
by  dividing  by  six;  it  will  be  noted  that  the  remainder  from  the 
degrees  gives  the  first  figure  of  the  minutes,  and  the  remainder 
from  the  minutes  gives  the  first  figure  of  the  seconds,  so  it 
becomes  unnecessary  to  reduce  these  remainders  to  the  next 
lower  unit,  as  would  be  required  with  any  other  number  of 
repetitions.  For  example,  let  the  accumulated  reading  be 
250°  57'  15", 

6)250°    57'  15" 
41     49    32.5' 

6  into  250  goes  41  times  and  4  over,  and  4  is  the  first  figure  of 
the  minutes ;  6  into  57  goes  9  times  and  3  over,  and  3  is  the  first 
figure  of  the  seconds. 

Second.  The  same  numerical  result  can  be  reached  without 
carrying  out  the  reduction  exactly  as  described  in  the  explanation 
of  the  method. 

Let  a  =  mean  of  verniers  at  beginning  of  a  set; 

b  =  mean  of  verniers  after  six  repetitions  on  interior 

angle; 
c  =  mean  of  verniers  after  six  repetitions  on  exterior 

angle; 
n  =  number  of  times  vernier  passes  initial  point  in  the 

six  repetitions  of  the  interior  angle; 
I  =  interior  angle  as  measured; 
E  =  exterior  angle  as  measured; 

v  =  adjustment  to  be  added  to  either  angle  as  measured; 
A  =  ad  justed  value  of  interior  angle. 

Since  the  interior  and  exterior  angles  together  make  360°, 
and  each  has  been  repeated  six  times,  the  total  angle  turned 
through  must  be  360°  X  6,  or  what  amounts  to  the  same  thing, 
5  complete  circuits  plus  the  indications  of  the  verniers  and  the 
correction  for  the  accumulated  errors;  so  that  if  n  equals  the 
number  of  complete  circuits  involved  in  the  six  repetitions  of 
the  interior  angle,  then  (5  —  n)  must  represent  the  number  of 
complete  circuits  involved  in  the  six  repetitions  of  the  exterior 
angle.  Hence 


MEASUREMENT  OF  ANGLES 


57 


E 


1  +  E  = 


T       360n  +  b  -  a 

6    -       ' 

360(5  -n)  +  c  -b 


360  X  5  +  c  -  a 


v  =^(360  - 


6 

360  X 


-  a\    _  1/360  -  c  +  a 
~2\       ~Q~ 


A  =  I  +  v, 

360n  +  b  -  a    ,    1/360  -  c  +  a 


6 


6 


l/360n  +  b  -  a       360n  +  (360  +  b)  -  c 
2\  6  ~6~ 


In  actual  work  no  attempt  is  made  to  observe  the  value  of  n, 
as  its  value  is  always  evident  from  the  approximate  value  of 
the  angle  as  given  by  the  first  reading.  The  remainder  of  the 
formula  involves  very  simple  operations  on  the  three  mean  vernier 
readings. 

40b.  Illustrative  Example.  A  complete  example  of  notes  and  reduc- 
tions for  a  double  set  of  angle  measurements  is  here  given  to  illustrate  the 
above  method. 


Station  occupied  =  A. 
Date  =  Aug.  28,  1911. 
Time  =  4.30  P.M. 

Angle  =  Sta.  B  to  Sta.  C. 
Observer  =  J.  H.  Smith. 
Instrument  =  Brandis  No.  17. 

Telescope. 

Ver.  A. 

Ver.  B. 

Mean. 

Angle. 

Average. 

— 

0°  00'  00" 

180°  00'  10" 

0°  00'  05" 

1.  D 

75   12    30 

6.  D&R 

91    14   50 

271    14  50 

91    14  50 

75°  12'  27".5 

6.  R&D 

359   58  50 

179   59  00 

359   58   55 

75    12    39  .2 

75°  12'  33".4 

6.  D&R 

91    13   50 

271    13  50 

91    13  50 

75    12    29  .2 

6.  R&D 

359   57   50 

179   57   50 

359    57   50 

75    12    40  .0 

75   12  34  .6 

Mean  angle  = 

75°  12'  34"  0 

58  GEODETIC  SURVEYING 

It  will  be  noted  that  vernier  A  was  set  to  zero  to  begin  with,  and  vernier 
B  read  180°  00'  10".  This  setting  to  zero  is,  of  course,  not  essential,  but 
convenient,  as  the  next  reading  at  once  gives  a  close  value  of  the  desired 
angle  without  computation.  There  is  no  object  in  reading  vernier  B  for 
this  approximate  determination.  The  remaining  readings  are  taken  at  the 
proper  time  just  as  the  instrument  reads,  paying  no  attention  to  the  number 
of  times  the  360°  point  has  been  passed.  1.  D  means  one  measurement  of 
the  angle  with  the  telescope  direct.  6.  D  &  R  means  six  repetitions,  using 
the  telescope  equally  both  direct  and  reversed  (hence  6.  D  &  R  means  the 
result  after  3  direct  and  3  reversed  measurements).  It  will  also  be  noted 
that  no  resetting  of  verniers  has  taken  place  at  any  time  throughout  the 
complete  double  set.  Vernier  B  is  only  read  in  order  to  average  out  instru- 
mental errors  (which  are  always  very  small),  and  therefore  in  filling  in  this 
column  the  degrees  are  recorded  the  same  as  given  by  vernier  A,  that 
is,  the  constant  difference  of  180°  between  vernier  A  and  B  is  not  allowed 
to  affect  the  mean.  In  filling  out  the  column  marked  angle  the  first  and 
the  final  reading  of  each  set  are  subtracted  from  the  middle  reading  (adding 
360°  if  necessary  to  make  the  subtraction  possible),  dividing  the  remainder 
by  6,  and  adding  as  many  times  60°  as  may  be  needed  to  make  the  result 
correspond  to  the  1.  D  reading. 

91°     14'    50"  91°    13'    50" 

0      00     05  360 


6)91      14     45  451      13      50 

15      12     27.5  359      58     55 


60  6)91       14      55 


75      12     27.5  15      12     29.2 

60 


75      12     29.2 

91°    14'    50"  91°    13'    50" 

360  360 


451       14      50  451       13      50 

359      58      55  359      57      50 


6)91       15      55  6)91       16      00 

15      12      39.2  15      12      40.0 

60  60 


75      12     39.2  75      12     40.0 

75      12     27.5  75      12     29.2 

75      12     39.2  75      12     40.0 


2)66.7 
33.4  34.6 

In  actual  practice  the  360°  and  the  60°  would  have  been  added  mentally  as 
needed. 

40c.  Additional  Instructions.  If  it  is  desired  to  attempt  to 
eliminate  errors  of  graduation,  several  double  sets  may  be  taken  at 
different  parts  of  the  circle,  symmetrically  disposed.  Modern 


MEASUREMENT  OF  ANGLES  59 

instruments  are  so  well  graduated,  however,  that  it  is  doubtful 
if  any  increased  accuracy  is  gained  by  this  refinement  when 
measuring  angles  by  any  method  of  repetition. 

If  it  is  desired  to  measure  more  than  one  angle  at  the  same 
station,  as  for  instance  A  OB  and  BOC,  Fig.  20,  we  may  take  six 
repetitions  on  each  of  these  angles  and  close  the  horizon  by  six 
repetitions  on  the  angle  from  C  clockwise  around  to  A,  and  divide 
the  failure  to  total  360°  equally  among  the  three  angles;  or  we 
may  measure  A  OB  and  its  exterior  angle  without  regard  to  station 
C,  and  then  measure  BOC  and  its  exterior 
angle  without  regard  to  station  A. 

In  using  the  above  or  any  other  methods 
of  measuring  an  angle  by  repetition  it  is 
presumed  the  surveyor  will  use  every  pre- 
caution possible  in  the  handling  of  the  instru- 
ment. Avoid  walking  around  the  instrument, 
if  supported  on  a  tripod;  un clamp  the  lower 
motion  and  revolve  the  instrument  if  it  is 
desired  to  read  the  verniers.  Do  not  relevel 
during  the  progress  of  measuring  an  angle  jiIG.  20. 

except  at  such  times  as  the  upper  motion  is 
clamped  and  the  lower  motion   free.      Revolve   the  instrument 
very  carefully  on  its  vertical   axis  to  avoid   slipping  the  plates. 
Read  each  vernier  independently,   without  regard   to  what  the 
other  one   may  have  read. 

41.  Adjustments  of  the  Repeating  Instrument.  For  the 
measurement  of  horizontal  angles  the  required  adjustments 
include : 

The  plate-bubble  adjustment; 
The  st riding-level  adjustment; 
The  collimation  adjustment; 
The  horizontal-axis  adjustment. 

These  adjustments  may  be  made  as  here  described,  but  there 
is  usually  more  than  one  way  of  making  the  same  adjustment. 

The  Plate-bubble  Adjustment.  This  is  made  in  the  same 
manner  as  with  a  surveyor's  transit.  Place  one  bubble  parallel 
to  two  of  the  leveling  screws,  and  bring  both  bubbles  to  the  center. 
Turn  the  instrument  180°  on  the  vertical  axis,  and  adjust  each 
bubble  for  one-half  of  its  movement.  Level  up  and  test  again, 


60  GEODETIC  SURVEYING 

and  so  continue  until  revolution  on  the  vertical  axis  causes  no 
movement  of  the  bubbles. 

The  Striding-level  Adjustment.  Level  up  the  instrument  by 
the  plate  bubbles  (not  absolutely  necessary  but  convenient). 
Place  striding  level  in  position  with  telescope  parallel  to  one  pair 
of  screws.  Bring  striding-level  bubble  to  center  with  remaining 
screw.  Lift  striding  level  off,  and  replace  in  reversed  position. 
Adjust  it  for  one-half  the  bubble  movement.  Again  bring  bubble 
to  middle  as  before  with  the  leveling  screw,  test  again,  and  repeat 
until  reversal  of  the  striding  level  causes  no  movement  of  its 
bubble. 

The  Collimation  Adjustment.  This  is  the  same  as  with  a 
surveyor's  transit.  Set  up  on  nearly  level  ground,  level  up 
with  the  plate  bubbles,  and  then  perfect  the  leveling  with  the 
striding  level,  so  that  revolution  on  the  vertical  axis  of  the 
instrument  causes  no  movement  of  the  striding-level  bubble. 
Unless  the  horizontal  axis  is  in  adjustment  this  stationary  posi- 
tion of  the  bubble  will  not  be  in  the  middle.  With  the  instrument 
clamped  set  a  point  about  200  feet  away,  plunge  and  set  a  second 
point  about  the  same  distance  in  the  opposite  direction,  with 
the  telescope  reversed.  Un clamp,  revolve  on  vertical  axis,  set 
on  first  point  with  telescope  reversed.  Plunge  and  set  a  third 
point  near  the  second  point.  Adjust  by  bringing  the  vertical 
hair  back  one  quarter  of  the  disagreement.  Repeat  the  whole 
process  until  no  discrepancy  can  be  detected. 

The  Horizontal-axis  Adjustment.  This  is  the  same  as  with  the 
surveyor's  transit.  Level  up  perfectly  with  the  striding  level 
near  an  approximately  vertical  wall  or  equivalent.  Set  on  a 
high  point,  with  instrument  clamped.  Drop  the  telescope  and 
mark  a  low  point  about  level  with  the  telescope.  Unclamp, 
revolve  on  vertical  axis,  and  set  on  high  point  with  the  telescope 
reversed.  .Drop  the  telescope  and  set  a  low  point  abreast  of  the 
first  low  point.  Adjust  the  horizontal  axis  so  that  the  line  of 
sight  will  pass  through  the  high  point  and  bisect  the  space  between 
the  low  points.  If  the  striding  level  and  the  horizontal  axis  are- 
both  in  adjustment  and  the  instrument  level,  the  striding-level 
bubble  should  stay  unmoved  in  its  middle  position  while  the 
instrument  is  turned  completely  around  on  its  vertical  axis. 

42.  The  Direction  Instrument  and  its  Use.  Besides  the 
features  common  to  all  first-class  instruments,  as  described  in 


MEASUREMENT  OF  ANGLES  61 

Art.  37,  the  direction  instrument  has  two  distinguishing  features: 
First,  it  has  only  one  vertical  axis,  so  that  angles  can  not  be  meas- 
ured by  repetition  (means  often  provided  for  shifting  the  limb 
between  sets  of  readings  must  not  be  used  for  angle  repetition); 
Second,  it  is  provided  with  two  or  more  micrometer  microscopes 
for  reading  the  angles  measured.  The  single  center  and  clamp, 
instead  of  the  two  centers  and  clamps  of  the  repeating  instru- 
ment, undoubtedly  add  to  the  stability  of  the  instrument  and 
the  trueness  of  its  motion.  The  limb  of  a  10-inch  or  12-inch 
direction  instrument  is  commonly  graduated  into  5-minute  spaces, 
and  the  micrometer  microscopes  enable  an  angle  to  be  read  at 
once  to  the  nearest  second,  as  described  later  on. 

In  using  the  direction  instrument  each  angle  is  read  a  number 
of  times,  and  the  results  averaged,  to  eliminate  errors  of  pointing; 
all  the  microscopes  are  read  at  each  pointing,  to  eliminate  eccen- 
tricity of  vertical  axis  or  microscopes ;  half  of  the  readings  are 
taken  with  the  telescope  direct  and  half  with  it  reversed,  to 
eliminate  errors  of  collimation  and  horizontal  axis;  half  of  the 
readings  are  taken  to  the  right  and  half  to  the  left,  to  eliminate 
errors  due  to  twisting  of  the  instrument  and  station.  In  the 
highest  grade  of  work  the  limb  of  the  instrument  is  shifted  between 
each  set  of  readings  through  an  angular  distance  equal  to  the 
angular  distance  between  the  successive  microscopes  divided  by 
the  number  of  sets,  to  eliminate  errors  of  graduation.  This  last 
refinement  may  be  omitted  in  ordinary  work. 

43.  First  Method  with  Direction  Instrument.  The  instru- 
ment having  been  set  up  and  leveled  with  the  telescope  in  its 
normal  position  is  directed  to  the  first  station,  and  all  of  the 
micrometers  read,  and  so  on  to  the  right  (clockwise)  to  each 
station  in  order,  the  values  of  the  different  angles  being  obtained 
by  taking  the  differences  of  the  successive  readings,  as  will  be 
illustrated  by  an  example  when  the  method  of  using  the  microm- 
eters is  explained.  When  the  last  station  to  the  right  has  been 
reached  the  instrument  may  be  turned  still  further  in  the  same 
direction  until  it  reaches  the  initial  station,  called  closing  the 
horizon,  and  any  difference  between  the  initial  and  final  readings 
equally  divided  among  all  the  angles,  but  experience  does  not 
appear  to  show  any  advantage  in  thus  closing  the  horizon,  and 
it  is  commonly  not  done.  When  the  last  pointing  to  the  right 
has  been  made,  the  instrument  is  brought  back  station  by  station 


62 


GEODETIC  SURVEYING 


to  the  initial  point,  thus  making  a  new  series  of  values  for  the 
angles.  The  right  and  left  pointings  are  again  repeated,  this 
time  with  the  telescope  reversed.  The  four  series  of  values  thus 
obtained  constitute  one  set,  and  as  many  sets  as  desired  may  be 
averaged  together.  When  for  any  cause  a  set  is  incomplete  or 
inconsistent  the  entire  set  is  rejected.  When  there  are  several 
angles  to  be  measured  at  one  station  they  are  sometimes  measured 
in  various  combinations  as  well  as  singly,  the  method  of  adjust- 


ment appearing  later.     The  program  in  measuring  a  single  angle, 
Fig.  21,  is  as  follows: 

PROGRAM 
FIRST  SET. 

1.  Level  the  instrument. 
Set  telescope  direct  and 

2,  Set  on  A  and  read  micrometers. 


3, 

4. 


n 
A 


Reverse  telescope  and 

5.  Set  on  A  and  read  micrometers. 

6.  "      B         "  " 

7.  "      A 

SECOND  SET. 

1.  Shift  limb.     Relevel. 
Leave  telescope  reversed  and 

2.  Set  on  A  and  read  micrometers. 


MEASUREMENT  OF  ANGLES  63 

Set  telescope  direct  and 

5.  Set  on  "A  and  read  micrometers. 

6.  "      B         "  " 

7.  "     A 

If  there  were  two  angles  to  be  measured  at  a  station,  as  illus- 
trated in  Fig.  22,  the  program  would  be  as  follows : 

PROGRAM 

FIRST  SET. 

1.  Level  the  instrument. 
Set  telescope  direct  and 

2.  Set  on  A  and  read  micrometers. 

3.  "      B         "  " 

4.  "      C 

5.  "     B 

6.  "     A 

Reverse  telescope  and 

7.  Set  on  A  and  read  micrometers. 

Q  li          R  ll  il 

o.  O 

Q  "  C  "  " 

10.  "     B 

11.  "     A 

SECOND  SET. 

1.  Shift  limb.     Relevel. 
Leave  telescope  reversed  and 

2.  Set  on  A  and  read  micrometers. 

3.  "     B         "  i( 

4.  *u      C 

5.  "     B 

6.  "     A 

Set  telescope  direct  and 

7.  Set  on  A  and  read  micrometers. 

8.  "      B         "  " 

9.  "      C 

10.  "     B 

11.  «     A 


64  GEODETIC  SURVEYING 

and  similarly  for  any  number  of  angles  at  one  station.  It  will  be 
noted  that  in  the  above  method  the  telescope  is  reversed  in  posi- 
tion only  at  the  initial  station. 

44.  Second  Method  with  Direction  Instrument.  If  it  is  not 
desired  to  make  so  many  pointings  (in  order  to  reduce  the  labor  and 
time)  the  telescope  may  be  reversed  at  both  the  initial  and  final 
stations  and  the  number  of  pointings  be  greatly  reduced. 
The  determination  of  the  different  angles,  however,  by  this  second 
method  would  not  be  considered  as  good  on  account  of  the  decreased 
number  of  pointings.  If  a  sufficient  number  of  sets  were  taken 
to  equalize  the  number  of  pointings  the  two  methods  would,  of 
course,  be  equivalent.  Referring  to  Fig.  21,  page  62,  the  program 
for  a  single  angle  by  the  second  method  would  be  as  follows : 

PROGRAM 

FIRST  SET. 

1.  Level  the  instrument. 

Set  telescope  direct  and 

2.  Set  on  A  and  read  micrometers. 

3.  "      B          "  " 

Reverse  telescope  and 

4.  Set  on  B  and  read  micrometers. 

5.  "     A 

SECOND  SET. 

1.  Shift  limb.     Relevel. 

Leave  telescope  reversed  and 

2.  Set  on  A  and  read  micrometers. 

3.  "     B         "  " 

Set  telescope  direct  and 

4.  Set  on  B  and  read  micrometers. 

5.  "     A 

Referring  to  Fig.  22,  page  62,  the  program  by  the  second 
method  for  two  angles  at  a  station  would  be  as  follows : 


MEASUREMENT  OF  ANGLES  65 

PROGRAM 

FIRST  SET. 

1.  Level  up  instrument. 
Set  telescope  direct  and 

2.  Set  on  A  and  read  micrometers. 

3.  "     B 

4.  "      C 

Reverse  telescope  and 

5.  Set  on  C  and  read  micrometers. 

6.  "      B 

7.  "     A 

SECOND  SET. 

1.  Shift  limb.     Relevel. 
Leave  telescope  reversed  and 

2.  Set  on  A  and  read  micrometers. 

3.  "     B 

4.  "      C          "  " 

Set  telescope  direct  and 

5.  Set  on  C  and  read  micrometers. 

6.  "      B         "  " 

7.  "     A 

and  similarly  for  any  number  of  angles  at  a  station. 

45.  The  Micrometer  Microscopes.  When  the  direction  instru- 
ment is  set  up  and  leveled  at  a  station  the  graduated  plate  occupies 
a  fixed  position  for  the  time  being.  The  framework  which 
supports  the  telescope  carries  two  or  more  microscopes  symmetric- 
ally disposed  around  the  center  of  the  instrument  and  focussed 
directly  on  the  graduated  ring.  As  the  telescope  is  swung  around 
from  station  to  station  the  zero  point  of  each  microscope  passes 
over  the  graduations  an  equal  angular  amount.  If  the  exact 
position  (on  the  graduated  ring)  of  the  zero  point  of  any  one  of  the 
microscopes  is  noted  for  two  different  stations,  then  the  difference 
of  these  readings  gives  the  angle  through  which  the  instrument 
has  been  turned,  and  consequently  the  angle  between  these 


66 


GEODETIC  SURVEYING 


stations.  Combined  with  each  microscope  is  an  instrument  called 
a  filar  micrometer,  by  means  of  which  the  exact  position  of  the 
zero  point  of  the  microscope  on  the  scale  may  be  determined. 

Fig.  23  represents  diagrammatically  a  sectional  view  of  a  filar 
micrometer.     A  is  the  micrometer  box,  attached  to  the  microscope 


FIG.  23. — Filar  Micrometer, 

as  seen  in  Fig.  17.     The  micrometer  is  made  up  of  the  following 
parts: 

A,  micrometer  box; 

b,  b,  fixed  guide  rods; 

c,  movable  frame  carrying  comb  scale  d\ 

d,  comb  scale  attached  to  movable  frame  c; 

e,  movable  frame  carrying  cross-hairs  /; 

/,  cross-hairs  attached  to  movable  frame  e; 

9)  9)  9)  spiral  springs  to  take  up  lost  motion  cf  movable 

frames  c  and  e; 

hj  fixed  screw  whose  revolution  adjusts  movable  frame  c; 
m,  micrometer  screw  attached  to  movable  frame  e; 
n,   fixed   nut   whose   revolution   moves   cross-hairs   across 

field  of  view; 

p,  milled  head  for  revolving  nut  n] 
s,  graduated  head  for  indicating  fractional    revolutions    of 

nut  n; 

t,  fixed  index  for  reading  scale  on  graduated  head  s; 
v,  dust  cap  to  protect  micrometer  screw  m. 

The  central  notch  of  the  comb  scale  is  marked  by  a  small  hole 
drilled  behind  it  (or  greater  depth  to  that  notch,  or  other  equiv- 
alent), and  is  intended  to  be  practically  at  the  center  of  the  field 


MEASUREMENT  OF  ANGLES  67 

t 

of  view.  Every  fifth  notch  i&  indicated  usually  by  its  greater 
depth  and  square  bottom.  All  counting  is  done  with  the  notches 
and  not  with  the  points  of  the  teeth.  Each  revolution  of  the 
micrometer  screw  moves  the  cross-hairs  over  a  space  equal  to  the 
distance  between  the  bottoms  of  two  adjacent  notches.  When 
the  microscope  is  properly  adjusted  the  image  of  the  graduated 
ring  is  formed  in  the  plane  of  the  micrometer  cross-hairs,  so  that 
both  image  and  cross-hairs  are  seen  sharply  defined  on  looking 
into  the  eyepiece,  the  microscope  ordinarily  having  a  magnifying 
power  of  30  to  50  diameters.  The  comb  scale  is  placed  as  close  as 
possible  to  the  cross-hairs  without  touching  them,  and  hence  is 
seen  at  the  same  time  and  in  sufficiently  good  focus.  As  ordinarily 
arranged  the  limb  of  the  instrument  is  graduated  into  five-minute 
spaces,  and  the  micrometer  head  into  sixty  spaces,  and  five 
revolutions  of  the  micrometer  screw  carry  the  cross-hairs  across 
the  image  of  the  limb  from  one  five-minute  division  to  the  next 
five-minute  division;  so  that  one  notch  on  the  comb  scale  or 
one  revolution  of  the  micrometer  screw  indicates  one  minute  of 
angle,  and  each  division  on  the  head  indicates  one  second. 

46.  Reading  the  Micrometers.  The  cross-hairs  /,  Fig.  23, 
consist  of  two  parallel  spider  threads,  placed  just  a  little  further 
apart  than  the  width  of  the  graduation  lines  on  the  instrument, 
so  that  when  a  graduation  line  comes  central  between  the  hairs 
a  narrow  illuminated  line  appears  to  lie  on  each  side  of  the  gradu- 
ation. It  is  found  in  practice  that  the  hairs  can  be  centered  over 
a  graduation  in  this  way  better  than  by  any  other  plan  (such 
as  a  single  thread  or  intersecting  threads).  Everything  being 
in  good  adjustment  the  zero  point  of  the  microscopes  is  the 
center  of  the  space  between  the  cross-hairs  when  they  are  opposite 
the  central  notch  of  the  comb  scale  and  the  zero  of  the  head  is 
opposite  the  index  line.  It  is  important  to  note  that  the  comb 
scale  is  not  an  essential  part  of  a  micrometer,  but  simply  a  con- 
venience, enabling  the  observer  to  see  at  any  moment  how  many 
complete  revolutions  of  the  micrometer  screw  have  taken  place 
at  any  time  without  keeping  track  of  the  matter  while  the  screw 
is  being  turned;  no  attempt  must  be  made  to  get  the  value  of  a 
reading  by  the  comb  scale  beyond  its  intended  purpose  of  indicat- 
ing whole  revolutions  or  single  minutes,  the  seconds  being  read 
entirely  from  the  micrometer  head;  as  long  as  the  comb  scale 
serves  its  intended  purpose,  therefore,  of  counting  whole  revolu- 


68 


GEODETIC  SURVEYING 


tions,  it  does  not  matter  whether  its  position  in  the  field  of  view 
is  microscopically  exact  or  not.  Referring  to  Fig.  24,  a  greatly 
exaggerated  view  is  given  of  what  is  seen  through  one  of  the 
microscopes  for  a  certain  pointing  of  the  telescope.  As  seen  in 
the  microscope  (which  reverses  the  actual  fact)  the  scale  reads 
from  left  to  right.  Assuming  the  cross-hairs  set  to  their  index 
or  zero  point  the  reading  is  seen  to  be  65°  10'  plus  the  value 
between  the  10-minute  division  and  the  center  between  the 
two  hairs.  Running  the  micrometer  screw  backwards  until 
the  hairs  exactly  center  over  the  10-minute  division  it  is  found 


FIG.  24. 

that  one  notch  has  been  passed  over  by  the  hairs,  but  that  they 
have  not  gone  far  enough  to  center  over  the  second  notch  from 
the  middle  one.  The  screw  has  therefore  been  turned  through 
more  than  one  but  less  than  two  revolutions.  The  numbers 
on  the  micrometer  head  increase  as  the  hairs  run  toward  the  left, 
and  assuming  the  index  to  stand  opposite  25  the  complete  microm- 
eter reading  is  1'  25.0",  making  the  complete  reading  for  the 
pointing 

65°  10'  +  1'  25.0"  =  65°  11'  25.0", 

if  no  corrections  were  required.  The  head  reading  is  usually 
estimated  to  the  nearest  tenth  of  a  second. 

46a.  Run   of   the  Micrometer.     It  is  not  found  practicable 
in  actual  work  to  adjust  the  microscopes  so  perfectly  that  the 


MEASUREMENT  OF  ANGLES  69 

screw  will  be  turned  through  exactly  five  revolutions  (or  indicate 
exactly  300  seconds)  -in  drawing  the  hairs  from  one  five-minute 
division  to  the  next  one,  but  the  excess  or  deficiency  should  not 
exceed  about  2".  The  closer  the  microscope  is  brought  to  the 
graduated  ring  the  larger  becomes  the  image  in  the  plane  of  the 
cross-hairs.  It  is  almost  impossible  to  get  the  image  the  exact 
size  that  corresponds  to  precisely  five  revolutions  of  the  screw; 
even  if  this  result  were  accomplished  at  one  part  of  the  graduated 
ring  the  instrument  is  seldom  made  so  true  that  it  would  hold 
good  all  around  the  ring,  either  on  account  of  slight  errors  in  the 
graduations  or  a  lack  of  perfect  trueness  of  the  ring  itself,  and 
many  other  reasons;  owing  to  temperature  changes  and  other 
reasons  it  will  not  remain  true  or  the  same  at  the  same  part  of 
the  ring.  In  running  from  one  scale  division  to  another  the  amount 
by  which  the  micrometer  measurement  varies  from  300  seconds 
is  called  the  run  of  the  micrometer  between  those  divisions,  and 
must  be  determined  at  the  time  the  pointing  is  made.  Whatever 
the  micrometer  head  may  read  when  the  hairs  are  set  over  one  five- 
minute  division,  it  must  necessarily  read  the  same  when  the  hairs 
are  advanced  to  the  next  five-minute  division,  provided  there  is 
no  run  of  the  micrometer,  that  is,  provided  that  the  screw  turns 
through  precisely  five  revolutions  or  300".  If  the  two  head  read- 
ings are  not  the  same  the  difference  gives  the  value  of  the  run  of 
the  micrometer  between  these  two  divisions.  In  drawing  the 
hairs  from  left  to  right  the  head  readings  decrease,  so  that  the 
micrometer  overruns  when  the  forward  head  reading  is  less 
than  the  backward  head  reading,  and  vice  versa.  The  run  of  the 
micrometer  for  the  300"  space,  therefore,  equals  the  backward 
head  reading  minus  the  forward  head  reading,  and  the  micrometer 
measurement  of  the  300"  space  equals  300"  plus  the  run  of  the 
micrometer  for  this  space.  Since  the  micrometer  does  not 
measure  the  five-minute  (300")  space  correctly,  it  follows  that  a 
proportionate  error  exists  for  intermediate  points;  or  for  any 
intermediate  point  we  have 

Correction  for  run 


Run  of  micrometer  for  300"  space 


Micrometer  measurement  for  intermediate  point 
Micrometer  measurement  of  300"  space 


70  GEODETIC  SUKVEYING 

Let    n  =  number  of  full  turns  to  back  division  ; 
o  =  back  head  reading; 
p  =  forward  head  reading; 
b  =  backward  reading  in  seconds  =  60  n  +  o; 
/  =  forward  reading  (so  called)  in  seconds  =  60  n  +  p; 


m 


d  =  run  of  micrometer  for  300  "  space  =  o  —  p  =  b  —  /; 
c  =  correction  for  run  to  value  b; 
D  =  300"; 
A  =  micrometer  measurement  of  300  "  space  =  300  +  d  = 

D  +  d; 

M  =  adjusted  micrometer  reading  to  add  to  scale  read- 
ing =  b  -  c; 

then 


~  D  +  d' 
db 


M  =b  - 


D  +  d' 
db 


D  +  d' 
substituting 


but  since  d  is  always  very  small  in  comparison  with  D  we  may 
write  instead  the  extremely  close  approximation. 

TIT  ,   d       md 

M  =  m  +  -  -  -  -p-, 

in  which  care  must  be  taken  to  use  d  algebraically  with  its  correct 
sign.  Since  the  adjusted  reading  is  based  entirely  on  b  and  /  it 
is  evidently  unnecessary  to  set  the  micrometer  to  its  zero  point 
before  reading  either  6  or  /.  When  a  pointing  is  made  in  actual 
work  b  is  taken  as  the  mean  value  of  all  the  back  readings  of  the 


MEASUREMENT  OF  ANGLES  71 

i 

different  microscopes,  and  /  a$  the  corresponding  mean  of  the 
forward  readings,  and~only  one  reduction  is  made  for  a  pointing. 
The  scale  reading  is  taken  for  one  micrometer  only.  The  eyepiece 
of  each  microscope  must  be  very  carefully  focussed  by  the  observer, 
as  any  perceptible  parallax  renders  good  work  impossible. 

A  complete  example  of  notes  and  reduction  is  given  on  pages 
72  and  73. 

47.  Adjustments  of  the  Direction  Instrument.  For  the 
measurement  of  horizontal  angles  the  required  adjustments 
include : 

The  plate-bubble  adjustment; 

The  st riding-level  adjustment; 

The  collimation  adjustment; 

The  horizontal -axis  adjustment; 

The  microscope  and  micrometer  adjustment. 

These  may  be  made  as  here  described,  but  there  is  usually 
more  than  one  way  of  making  the  same  adjustment. 

The  Plate-bubble  Adjustment.  This  is  made  in  the  same 
manner  as  with  a  surveyor's  transit.  Place  one  bubble  parallel 
to  two  of  the  leveling  screws,  and  bring  both  bubble's  to  the 
center.  Turn  the  instrument  180°  on  the  vertical  axis,  and 
adjust  each  bubble  for  one-half  its  movement.  Level  up  and 
test  again,  and  so  continue  until  revolution  on  the  vertical  axis 
causes  no  movement  of  the  bubbles. 

The  Striding-level  Adjustment.  Level  up  the  instrument  by 
the  plate  bubbles  (not  absolutely  necessary  but  convenient). 
Place  striding  level  in  position  with  telescope  parallel  to  one  pair 
of  screws.  Bring  striding-level  bubble  to  center  with  remaining 
screw.  Lift  striding  level  off,  and  replace  in  reversed  position. 
Adjust  it  for  one-half  the  bubble  movement.  Again  bring  bubble 
to  middle  as  before  with  the  leveling  screw,  test  again,  and  repeat 
until  reversal  of  striding  level  causes  no  movement  of  its  bubble. 

The  Collimation  Adjustment.  This  is  the  same  as  with  a 
surveyor's  transit.  Set  up  on  nearly  level  ground,  level  up  with 
the  plate  bubbles,  and  then  perfect  the  leveling  with  the  strid- 
ing level,  so  that  revolution  on  the  vertical  axis  of  the  instrument 
causes  no  movement  of  the  striding-level  bubble.  Unless  the 
horizontal  axis  is  in  adjustment  this  stationary  position  of  the 
bubble  will  not  be  in  the  middle.  With  the  instrument  clamped, 


72 


GEODETIC  SUEVEYING 


ANGLE    MEASUREMENT    WITH 


Station  occupied  =  Sta.  A. 
Date  =  May  15,  1910. 
Time  =  5.00  P.M. 

Station 

Instru- 
ment. 

Microm- 
eter. 

Scale. 

b 

/ 

m 

B 

D 

A 

65°  10' 

85".0 

82".7 

B 

83  .4 

87  .6 

Mean 

84  .2 

85  .2 

84".  70 

C 

D 

A 

75°  15' 

126  .4 

124  .0 

B 

124  .2 

125  .2 

Mean 

125  .3 

124  .6 

124  .95 

C 

R 

A 

125  .2 

123  .1 

B 

123  .0 

126  .3 

Mean 

124  .1 

124  .7 

124  .40 

B 

R 

A 

82  .5 

80  .3 

B 

81   .0 

84  .6 

Mean 

81   .8 

82  .5 

82  .15 

LIMB  SHIFTED  ABOUT  90°. 

B 

R 

A 

72".6 

69".4 

B 

69  .8 

71   .8 

Mean 

71  .4 

70  .6 

71".00 

C 

R 

A 

112  .8 

110  .0 

B 

109  .8 

111  .6 

Mean 

111  .3 

110  .8 

111   .10 

C 

D 

A 

111   .5 

110  .1 

B 

109  .1 

111   .2 

Mean 

110  .3 

110  .7 

110  .50 

B 

D 

A 

70  .1 

69  .0 

B 

68  .0 

71   .2 

Mean 

T>9  .1 

70  .1 

69.  60 

MEASUREMENT  OF  ANGLES 


73 


THE    DIRECTION    INSTRUMENT 


Angle  =  Sta.  B  to  Sta.  C. 
Observer  =  Wm.  S.  Brown. 
Instrument  =  Branch's  No.  20. 

d     md 
2       D 

M 

Pointing. 

Angle. 

Average. 

-  0".22 

84".48 

65°  11'  24".48 

+  0  .06 

125  .01 

75    17    05  .01 

10°  05'  40".53 

-  0  .05 

124  .35 

75    17    04  .35 

10°  05'  41  ".45 

-  0  .16 

81   .99 

65    11    21   .99 

10    05    42  .36 

+  0  .21 

71   .21 

65    11    11   .21 

+  0  .06 

111   .16 

75    16    51   .16 

10    05    39  .95 

-  0  .05 

110  .45 

75    16    50  .45 

10    05    40  .54 

-  0  .27 

69  .33 

65     11  09  .33 

10    05    41   .12 

Mean  angle  = 

10°  05'  41  ".00 

74  GEODETIC  SURVEYING 

set  a  point  about  200  feet  away,  plunge  and  set  a  second  point 
in  the  opposite  direction  with  telescope  reversed.  Unclamp, 
revolve  on  vertical  axis,  set  on  first  point  with  telescope  reversed. 
Plunge  and  set  a  third  point  near  the  second  point.  Adjust  by 
bringing  the  vertical  hair  back  one  quarter  of  the  disagreement. 
Repeat  the  whole  process  until  no  discrepancy  can  be  detected. 

The  Horizontal-axis  Adjustment.  This  is  the  same  as  with  the 
surveyor's  transit.  Level  up  perfectly  with  the  striding  level  near 
an  approximately  vertical  wall  or  equivalent.  Set  on  a  high 
point,  with  instrument  clamped.  Drop  the  telescope  and  mark 
a  low  point  about  level  with  the  telescope.  Unclamp,  revolve 
on  vertical  axis,  and  set  on  high  point  with  the  telescope  reversed. 
Drop  the  telescope  and  set  a  low  point  abreast  of  the  first  low 
point.  Adjust  the  horizontal  axis  so  that  the  line  of  sight  will 
pass  through  the  high  point  and  bisect  the  space  between  the 
low  points.  If  the  striding  level  and  the  horizontal  axis  are  both 
in  adjustment  and  the  instrument  level,  the  striding-level  bubble 
should  stay  unmoved  in  its  middle  position  while  the  instrument 
is  turned  completely  around  on  its  vertical  axis. 

The  Microscope  and  Micrometer  Adjustment.  It  is  necessary 
to  have  the  graduated  arc  pass  practically  across  the  center  of 
the  field  of  view,  and  the  supporting  frame  is  generally  provided 
with  self-evident  means  of  making  this  adjustment.  Sometimes 
all  but  one  of  the  microscopes  may  be  moved  circumferentially 
so  as  to  space  them  equally  around  the  circle,  but  frequently 
they  are  permanently  mounted  by  the  makers  in  their  proper 
places.  The  microscope  tube  may  be  rotated  on  its  own  axis 
until  the  cross-hairs  are  exactly  parallel  to  the  graduation  lines. 
The  microscope  can  be  adjusted  so  as  to  change  the  distance 
between  the  objective  and  the  cross-hairs,  and  the  whole  micro- 
scope can  be  moved  so  as  to  change  the  distance  between  the 
objective  and  the  graduated  plate;  if  the  micrometer  overruns, 
the  image  of  the  graduations  is  too  large,  and  must  be  made 
smaller  by  decreasing  the  distance  between  the  objective  and 
cross-hairs  slightly,  and  then  carefully  moving  the  whole  micro- 
scope away  from  the  graduations  until  a  perfect  focus  is  again 
obtained  exactly  in  the  plane  of  the  cross-hairs,  as  shown  by  the 
fact  that  properly  focussing  the  eyepiece  shows  both  the  hairs 
and  the  graduations  sharply  defined  and  without  parallax;  if 
the  micrometer  underruns  the  image  is  too  small,  the  objective 


MEASUREMENT  OF  ANGLES  75 

must  be  moved  away  from  the  cross-hairs,  and  the  whole  micro- 
scope moved  toward  the  graduations;  this  adjustment  should  be 
perfected  until  the  error  does  not  exceed  2".  The  zero  point 
of  each  microscope  can  be  changed  by  shifting  "the  comb  scale 
and  revolving  the  graduated  head  on  the  micrometer  screw; 
this  adjustment  enables  two  microscopes  to  be  set  exactly  180° 
apart,  three  microscopes  120°  apart,  etc.  Great  care  and  skill 
are  necessary  to  properly  adjust  the  microscopes  and  micrometers. 
48.  Reduction  to  Center.  It  is  sometimes  impossible  to  set 
up  an  instrument  exactly  over  a  given  station,  a  flag  pole  or  steeple, 
for  instance.  In  such  a  case  an  eccentric  station  is  taken  as  near 
the  true  station  as  possible,  and  the  eccentric  angle  is  measured 
with  the  same  precision  as  would  have  been  used  for  the  real 
angle.  From  the  location  of  the  true  station  with  reference  to 
the  eccentric  station  a  correction  is  computed  which  will  reduce 
the  eccentric  angle  to  what  it  would  have  been  if  measured  at 
the  true  station,  this  operation  being  known  as  reduction  to  center. 
The  true  station  is  generally  referred  to  the  eccentric  station 
by  an  angle  and  a  distance,  a  single  measurement  of  the  angle 
being  sufficiently  accurate  for  the  pur- 
pose. Referring  to  Fig.  25,  C  is  the 
true  station,  E  the  eccentric  station, 
ACE  the  desired  angle,  AEB  the 
angle  actually  measured,  and  a  and  r 
the  angle  and  distance  connecting  the 
true  station  with  the  eccentric  station. 
In  the  triangle  ABC  the  angles  at  A 
and  B  are  known  by  actual  measure- 
ment, and  one  of  the  sides  of  the 
triangle  must  be  known  by  measure- 
ment or  by  computation  from  its  con- 
nection with  the  triangulation  system.  Having  one  side  and  two 
angles  given  we  may  regard  all  the  parts  of  the  triangle  ABC  as 
known  with  sufficient  accuracy  for  the  present  reduction,  on 
account  of  the  desired  correction  always  being  very  small.  Oppo- 
site angles  at  D  being  equal,  we  have 

C  +  y  =  E+  x^ 
or 

C  =  E  +  (x  -  y), 


76 
but 

hence 


GEODETIC  SURVEYING 

sin  z  r  ,     sin  y       r 

-. — T^P, r  =  r-     and     — — -  =  — • 

sin  (E  +  a)       b  sin  a 

r  sin  (E  +  a) 
sin  z  =  -         ,  and     sin  y 


a' 
r  sin  o: 


Since  x  and  y  are  very  small  angles,  we  may  write 


sin  x 


or 


=  x  sin  V     and     sin  y  =  y  sin  1", 


whence 


+  a)  /     r    \  sin  a 

ana     v  =  (  — — —r-. 

>  \  si  n    I ' '  /       n. 


r      Fsin  (E  +  a)       sin  al 
sinl/rL          &  "~a"J' 


in  which  the  correction  to  be  applied  to  E  will  be  in  seconds, 
and  may  be  essentially  positive  or  negative,  since  the  true  angle 

may  be  either  larger  or  smaller  than 
the  eccentric  angle.  If  care  is  taken 
to  use  the  proper  value  of  a,  to  re- 
member that  angles  between  180°  and 
360°  have  negative  sines,  and  to  work 
out  the  formula  for  C  algebraically,  the 
correct  value  of  C  will  be  obtained 
whether  it  be  larger  or  smaller  than  Et 
and  without  knowing  what  the  plotted 
figure  would  look  like.  If  measured 
from  r  the  angle  a  must  be  taken 
counter-clockwise  all  the  way  around  to 

the  line  EB  no  matter  how  large  it  may  come;  if  measured 
from  EB  it  must  be  taken  clockwise  around  to  r;  thus  in 
Fig.  26  the  angle  a  is  the  one  so  marked  and  not  the  insi.'e 
angle  BEC. 

The  correction  to  be  applied  to  E  to  obtain  C  depends  entirely 
on  the  values  of  x  and  y,  and  these  may  be  computed  directly 
if  preferred,  and  combined  in  the  proper  way  by  inspection  of 
the  figure,  since  the  observer  can  scarcely  be  ignorant  of  how  the 
different  stations  are  related  to  each  other  and  hence  can  quickly 


FIG.  26. 


MEASUREMENT  OF  ANGLES 


77 


o 


78  GEODETIC  SURVEYING 

draw  a  sketch  of  the  actual  conditions.     All   the   possible  cases 
are  shown  in  Fig.  27,  page  77,  for  any  angle  less  than  180°. 

49.  Eccentricity  of    Signal.     It  sometimes  becomes  necessary 
in  measuring  an  angle  to  sight  on  an  eccentric  signal ;  for  instance, 

as  in  Fig.  28,  it  may  be  necessary  to 
sight  to  B'  instead  of  the  true  station 
B.  The  measured  angle  ACS'  must 
therefore  be  corrected  by  the  small 
angle  BCB'  to  obtain  the  desired  angle 
ACB.  In  the  triangle  ABC  the  angles 
at  A  and  B  are  measured,  and  one  side 
is  always  known  through  connection 
with  the  rest  of  the  system,  so  that 
the  side  BC  can  be  computed  with 
sufficient  closeness  for  the  present 
purpose.  The  distance  BD,  perpen- 
dicular to  CB'  and  called  the  eccen- 
tricity, is  either  directly  measured  or  computed  from  a  measure- 
ment of  the  distance  B'B  and  the  angle  at  B'.  Then 

T)  7~\ 

BCB'  (in  seconds)  = 


BC  sin  1"' 

60.  Accuracy  of  Angle  Measurements.  When  the  same 
instrument  is  used  by  a  skilled  observer  under  the  same  condi- 
tions results  are  obtained  which  differ  but  slightly  from  each  other. 
In  measuring  an  angle  with  an  ordinary  30-second  transit  of 
good  make  two  sets  taken  by  the  method  of  repetition,  in  accord- 
ance with  the  example  given  on  page  57,  should  not  differ  by 
more  than  5".  A  10-inch  repeating  instrument  used  in  the  same 
way,  or  a  10-inch  direction  instrument  used  in  accordance  with  the 
example  on  pages  72  and  73,  should  give  sets  differing  by  less  than 
2".  A  great  many  sets  may  be  taken  at  the  same  time  and  agree 
with  each  other  within  these  limits,  but  it  does  not  follow  that  the 
value  of  the  angle  is  obtained  with  this  degree  of  precision.  If  the 
same  observer  measures  the  same  angle  with  the  same  instrument 
under  different  conditions  a  new  series  of  values  may  be  obtained 
closely  agreeing  with  each  other,  but  the  mean  of  the  values 
belonging  to  the  first  series  may  differ  several  seconds  from  the 
mean  of  the  second  series;  in  fact,  the  two  means  may  differ  more 
from  each  other  than  the  result  of  any  one  set  differs  from  the 


MEASUREMENT  OF  ANGLES  79 

mean  of  its  own  series.  Mo/ning  measurements  often  differ 
from  afternoon  measurements,  even  when  the  atmospheric 
conditions  appear  to  be  the  same.  In  the  finest  work  an  angle 
is  measured  on  many  different  days  (sometimes  with  an  equal 
number  of  A.M.  and  P.M.  measurements),  under  as  different 
conditions  as  possible,  and  a  general  average  taken  of  all  the  values 
obtained,  called  the  arithmetic  mean. 

In  the  Coast  Survey  work  the  probable  error  (Chapter  XIII) 
of  a  primary  angle  must  not  exceed  0".3,  and  primary  triangles 
must  close  within  3".  In  secondary  work  the  probable  error 
of  an  angle  must  not  exceed  0".7,  and  triangles  must  close 
within  6".  In  work  of  less  importance  a  greater  probable  error 
is  allowable,  but  the  triangles  are  expected  to  close  within  about 
12".  A  sufficient  number  of  measurements  must  be  taken  to 
bring  about  these  results,  but  in  primary  work  in  any  event 
at  least  five  double  sets  like  those  given  in  the  examples  ought 
to  be  taken. 

The  probable  error  of  an  angle  is  obtained  as  follows : 

Let  ra  =  probable  error  of  mean  angle  (in  seconds) ; 

MI,  M2,  etc.  =  value  given  by  each  set; 

z  =  mean  value  of  angle; 
Mi  —  z  = 


,,  ,  etc.  = residuals  (in  seconds) ; 

M2  -  z  =  v2  J 

2-y2  =  sum  of  squares  of  residuals; 
n  =  number  of  sets. 

then 


r  s^2 

\n(n  -  1)' 


ra=  ±  0.6745 


Example,     Six  measurements  of  an  angle  were  taken: 
Observed  Values.       Arithmetic  Mean.  v 


7° 

16'  9" 

.2 

7° 

16' 

9" 

.7 

— 

0" 

.5 

0 

.25 

7 

16  12 

.1 

7 

16 

9 

.7 

+ 

2 

.4 

5 

76 

7 

16   8 

.4 

7 

16 

9 

.7 

— 

1 

.3 

1 

69 

7 

16  6 

.7 

7 

16 

9 

.7 

— 

3 

0 

9. 

00 

7 

16  10 

.3 

7 

16 

9 

.7 

4- 

0 

.6 

0 

.36 

7 

16  11 

.5 

7 

16 

9 

.7 

-f 

1 

.8 

3 

.24 

6)58" 

.2 

0" 

.0 

20 

.30 

9 

.7 

80  GEODETIC  SURVEYING 

The  algebraic  sum  of  the  residuals  is  zero,  as  it  always  should  be. 


r«  =  ±  0.6745V  J^V±0'' 


6(6 

If  the  several  determinations  of  the  angle  are  not  considered 
equally  good  (on  account  of  a  difference  in  the  number  of  repe- 
titions or  in  the  atmospheric  conditions,  etc.),  and  the  values  are 
correspondingly  weighted,  each  value  is  multiplied  by  its  weight 
and  the  sum  of  the  products  divided  by  the  sum  of  the  weights, 
giving  the  weighted  arithmetic  mean,  the  probable  error  of  which  is 


4 


±0.6745 


in  which  Epv*  equals  the  sum  of  the  results  obtained  by  multiplying 
each  squared  residual  by  the  corresponding  weight ;  and  Sp  equals 
the  sum  of  the  weights. 


CHAPTER  IV 
TRIANGULATION   ADJUSTMENTS   AND   COMPUTATIONS 

51.  Adjustments.    After  the  field  work  of  angle  measurement 
has  been   completed   there   still   remains  the   office   adjustment 
of  the  angles  necessary  to  satisfy  the  rigid  geometrical  conditions 
involved;    thus  all  the  angles  around  a  point  must  add  up  to 
360°,  the  three  angles  of  a  triangle  must  add  up  to  180°,  etc. 
All   such  geometrical   conditions  must  be  satisfied    before   .the 
lengths  of  the  various  lines  of  the  system  are  computed.     The 
adjustment  of  the  angles  at  any  station  without  regard  to  meas- 
urements taken  at  other  stations   (such  as  making  the  angles 
around  a  point  add  up  to  360°) ,  is  called  station  adjustment.     The 
mutual  adjustment  of  the  several  angles  of  a  given  figure  (such 
as  making  the  angles  of  a  triangle  add  up  to  180°),  is  called  figure 
adjustment.     Easily  applied  rules  for  simple  cases  of  adjustment 
can  be  .derived  by  the  method  of  least  squares  or  the  theory  of 
weights;    more  complicated   cases   are  better  adjusted   directly 
by  the  method  of  least  squares,  as   explained  in  Part  II  of  this 
book.     The  object  in  any  case  of  adjustment  is,  of  course,  to 
find  from  the  measured  values  the  most  probable  values  con- 
sistent with  the  geometrical  conditions  involved. 

52.  Theory  of  Weights.    The  weight  of  a  quantity  is  defined 
as  its  relative  worth.     The  term  weight,  therefore,  is  purely  relative 
and  must  never  be  understood  in  an  absolute  sense.     A  distance  of 
3  feet  or  3  miles  is  an  absolute  and  definite  distance ;  a  weight  of  3 
does  not  represent  any  definite  degree  of  precision,  but  is  simply  a 
comparison  with  that  which  is  assigned  a  weight  of  1 .    The  basis  of 
comparison  is  fundamentally  the  number  of  observations  of  unit 
weight  from  which  the  given  value  is  derived ;  thus  if  5  measure- 
ments of  an  angle  were  regarded  as  equally  reliable,  expressed 
mathematically  by  assigning  to  each  a  weight  of  1,  the  mean  value 
of  the  angle  (by  definition)  would  have  a  weight  of  5.     Weights  are 
often  arbitrarily  assigned   as   a   matter   of  judgment,  however, 

81 


82  GEODETIC  SURVEYING 

where  the  corresponding  number  of  observations  does  not  exist; 
thus  a  measurement  obtained  under  unusually  favorable  condi- 
tions might  be  considered  as  good  as  the  mean  of  two  measure- 
ments taken  under  less  favorable  conditions,  and  hence  a  weight 
of  2  assigned  to  this  single  favorable  measurement.  Since, 
therefore,  the  numbers  representing  weight  are  purely  relative, 
and  do  not  necessarily  represent  a  corresponding  number  of 
observations,  any  number,  whole  or  fractional,  may  be  so  used; 
thus  two  quantities  may  be  said  to  have  the  weights  respectively 
of  1  and  2,  or  \  and  1,  or  0.12  and  0.24,  and  their  relative  worth 
is  the  same  in  either  case.  The  mean  value  as  understood  above 
is  the  arithmetic  mean,  and  is  only  used  when  the  quantities  are 
of  equal  weight.  When  the  different  values  are  of  unequal  weight 
each  value  is  multiplied  by  its  weight,  and  the  sum  of  the  products 
is  divided  by  the  sum  of  the  weights,  the  result  obtained  being 
called  the  weighted  arithmetic  mean. 

53.  Laws  of  Weights.    The  following  principles  (established 
by  the  method  of  least  squares)  govern  the  use  of  weights : 

1.  The  weight  of  the  arithmetic  mean   (with  measurements 
of  unit  weight)  equals  the  number  of  observations. 

Example.    Angle  A  by  different  mensurements  equals 

29°  21'  59".  1,  weight  1 

29    22    06   .4,  "1 

29    21    58   .3,  "       I 

3)88°  06'  03". 6 

Arithmetic  mean  =     29°  22'  01". 2,  weight  3. 

2.  The  weight  of  the  weighted  arithmetic  mean  equals  the  sum 
of  the  individual  weights. 

Example.    Base  line  AB  by  different  measurements  equals 

4863.241ft.,  weight  2 
4863.182ft.,      "       1 

whence 

4863.241  X2  =  9726.482 
4863.182  X  1_=  4863.182 
3)14589.664 
Weighted  arithmetic  mean=  4863.221,  weight  3. 


TRIANGULATION  ADJUSTMENTS  AND  COMPUTATIONS    83 

3.  The  weight  of  the  algebraic  sum  of  two  or  more  numbers  is 
equal  to  the  reciprocal  of  the  sum  of  the  reciprocals  of  the  indi- 
vidual weights. 

Example.     Angle  A  =  45°  14'  11"  .2,     weight  2 
Angle  B  =  11    21    19     .6,          "       3 

1  fi 

A  +  B    =56°  35'  30". 8,      weight  =—     -=  -; 

IT*  0 

A  -  B    =33°  52'  51". 6,       weight  = 


4.  Multiplying  a  quantity  by  a  factor  divides  its  weight  by 
the  square  of  that  factor. 

Example.     Angle  A  =    67°  10'  12". 5,     weight  3, 

3  3 


2A  =  134°  20'  25". 0,     weight 


2X2       4 


5.  Dividing  a  quantity  by  a  factor  multiplies  its  weight  by 
the  square  of  that  factor. 

Example.     Base  AB  =  2716. 124  ft.,     weight  3, 

A  7-? 

—  =  1358.062  ft.,     weight  =  3  X  4  =  12. 

6.  Multiplying  an  equation  by  its  own  weight  (or  dividing  it 
by  the  reciprocal  of  its  weight),  inverts  its  weight. 

Example,    f  (z  +  y)  =  400,  weight  f ;  multiplying  by  f  (or  dividing  by 
|),  we  have 

4 
2(s  +  y)  =  300,  weight   -. 

7.  Changing  all  the  signs  of  an  equation,  or  combining  the 
equation  with  a  constant  by  addition  or  subtraction,  leaves  the 
weight  unchanged. 

Example,     x  +  y  =    11°  10'  14". 6,     weight  2.3, 
and 

360°  _  (x  +  y)  =  348°  49'  45". 4,     weight  2.3. 


84 


GEODETIC  SURVEYING 


U 


54.  Station    Adjustment.      This     consists,    as     explained    in 
Art.  51,  of  making  the  angles  at  a  station  geometrically  consistent, 

such  as  making  all  the  angles  around 
a  point  add  up  to  360°.  The  following 
cases  are  worked  out  as  shown  : 

Case  1.  The  angles  at  a  point  have 
been  measured  with  equal  care  (giving 
them  equal  or  unit  weight).  In  this 
case  any  discrepancy  is  equally  dis- 
tributed among  the  three  angles.  Thus 
in  Fig.  29,  if  the  angles  x,  y,  z,  as 
measured,  added  up  to  360°  00'  06", 
then  each  measured  value  would  be  re- 
duced by  2". 

As  an  application  of  the  theory  of  weights,  let  us  suppose 
we  have  by  measurement 

x  =  a,     weight  1 


FIG.  29. 


z  =  c, 


From  third  observation  360°  —  z  =  360°  —  c,       weight  1 

or  x  +  y  =  360°  -  c,  "      1 

By  second  observation  y  =      b,  "      1 


By  subtraction 

By  first  observation 


x  =360°  -  b  -c,  weight^ 
x  =      a,  "      1 


Taking  the  weighted  arithmetic  mean  of  these  values  of  x, 


\x  =  i(360°  -  b  -  c) 

x  =  a 


By  addition 
whence 


\x  =  a  +  £(360°  -  b  -  c) 
x  =  fa  +  £(360°  -  b  -  c) 
=  a  +  J(360°  -  a  -  b  r  c) 
=  a  +  |  [360°  -  (a  +  b  +  c)], 


which  indicates  that  the  most  probable  value  of  x  is  found  by 
correcting  the  measured  value  a  by  one-third  the  discrepancy  ;  and 
the  same  result  would  be  reached  for  y  and  z.  In  combining  the 
observations  as  above  it  is  to  be  noted  that  each  observation  can 


TRIANGULATION  ADJUSTMENTS  AND  COMPUTATIONS    85 

be  used  but  once,  as  otherwise^  additional  observations  would  be 
implied  that  in  fact  Rave  not  been  taken.  The  above  rule  for 
the  distribution  of  the  station  error  is,  of  course,  the  same  as 
would  be  obtained  by  the  method  of  least  squares. 

Case  2.  The  angles  as  measured  around  a  point,  Fig.  29, 
have  been  assigned  different  weights.  In  this  case  any  discrep- 
ancy is  distributed  inversely  as  the  weights.  Thus  if  the  weights 
are 

for  x,  1,  for  y,  2,  for  z,  3, 

the  distribution  of  error  would  be  as 

I.I.I 
I  :  2  :  3 ' 
which  is  the  same  as 

6.3.2 

6  :  6  :  6' 
which  is  equivalent  to 

6:3:2; 

and  since  6+3+2  =  11,  we  have 

n 

correction  f or  x  =  —  of  discrepancy ; 

3 

correction  for  y  =  —  of  discrepancy; 

2 

correction  for  z  =  —  of  discrepancy. 

Case  3.  Several  angles  at  a  point,  and  also  their  sum,  have 
been  measured  with  equal  care.  In 
this  case  any  discrepancy  is  to  be 
equally  distributed  among  all  the  meas- 
ured values  (including  the  measured 
sum).  When  the  measured  sum  of 
several  angles  is  greater  than  the  sum 
of  the  individual  measurements,  the 
correction  is  positive  for  the  single 
measurements  and  negative  for  the 
measured  sum,  and  vice  versa.  Thus  FlQ 

in  Fig.  30,  if  the  entire  angle  measured 
8"  more  than  the  sum  of  the  single  measurements,  then  the  x,  y, 


86  GEODETIC  SURVEYING 

and  z  measurements  would  each  be  increased  by  2",  and  the 
measured  sum  would  be  reduced  by  2". 

Case  4.  Several  angles  at  a  point,  and  also  their  sum,  have 
been  measured,  and  different  weights  have  been  assigned  to  the 
measured  values.  In  this  case  any  discrepancy  is  distributed 
among  all  the  measured  values  inversely  as  their  weights.  Thus 
in  Fig.  30,  page  85,  suppose 

x  measured  with  weight  2; 

y       "  "        i; 

z         "  "  3; 

(x+y  +  z)       "  "          I, 

the  division  of  error  would  be  as 

!  -I    I    I 

2 :  i  :  3  :  r 

which  is  the  same  as 

362.6 
6  :  6  :  6  :  6' 

which  equals 

3:6:2:6; 

and  since  3  +  6  +  2  +  6  =  17,  we  have 

3 
correction  for  x  =  y=  of  discrepancy ; 

«  «  y  =  —  "  " 

17 

o 

U  ((  y     —    —    "  " 

~  17 
"  •  (x  +  y  +  z)  ==  A  « 

If  the  measured  values  of  x,  y,  and  z  add  up  to  less  than  the 
measured  sum  (x  +  y  +  z),  then  the  corrections  for  x,  y,  and  z, 
are  to  be  added,  and  the  correction  for  (x  +  y  +  z)  subtracted, 
and  vice  versa. 

General  Rule.  Any  case  of  station  adjustment  in  which  the 
coefficients  in  the  equations  are  all  unity  and  the  signs  are  all 


TEIANGULATION  ADJUSTMENTS  AND  COMPUTATIONS    87 


positive  (as  is  usually  the  case),  and  in  which  the  horizon  has  not 
been  closed  or  the  closing  has  been  .evaded  in  the  equations  by 
subtracting  one  or  more  angles  from  360°,  and  in  which  the  weights 
of  the  final  results  are  not  desired,  may  be  solved  as  follows: 
Multiply  each  equation  by  its  own  weight ;  add  together  separately 
all  the  new  equations  containing  x,  y,  z,  etc.,  as  shown  in  the 
following  example,  and  solve  the  resulting  equations  as  simulta- 


neous. 

Example.    Observed  values,  Fig.  31, 

x  =    14°  11'  17".l,  weight  1 

y  =    19    07  21  .3,       "      2 

x  +  y  =    33     18  43  .4, 

z  =  326    41  18  .2, 

y  +  z  =  345    48  39  .2, 

Subtracting  the  angles  involving  z  from  360°, 

x  =  14°  11'  17".l, 

y  =  19    07   21  .3, 

x  +  y  =  33     18    43  .4, 

360°  -  z  =  x  +  y  =  33     18    41  .8, 
360°  -  (y  +  z)  =  x  =  14     11    20  .8. 

Multiplying  each  equation  by  its  weight, 

x  =  14°  11'  17".  I 

+  2y  =  38  14  42  .6 

x  +  y  =  33  18  43  .4 

2x  +  2y  =  66  37  23  .6 

3x  =  42  34  02  .4 

Combining  separately  all  equations  containing  x,  and  all  equations  contain- 
ing y,  we  have 

7x  +  3y  =  156°  41'  26".5 
3x  +  5y  =  138    10   49  .6 

which,  solved  as  simultaneous  equations,  give 


FIG.  31. 


x  =    14' 
y  =    19 


11'  20". U 
07    21  .83 


the  sum  of  which  subtracted  from  360°  gives 

z  =  326°  41'  18".03. 

55.  Figure  Adjustment.  Having  found  by  measurement  and 
station  adjustment  the  best  attainable  values  of  the  different 
angles  of  a  system,  the  next  step  is  to  make  the  figure  adjustment. 
(If  the  work  is  very  important  and  the  angles  so  involved  that 


88  GEODETIC  SURVEYING 

making  the  figure  adjustment  would  disturb  the  station  adjust- 
ment, all  the  adjustments  would  have  to  be  made  in  one  operation 
by  the  method  of  least  squares.)  The  figure  adjustment,  as 
explained  in  Art.  51,  consists  in  making  such  slight  changes  in 
the  various  measured  angles  as  will  make  the  figure  geometrically 
consistent,  such  as  making  the  angles  of  a  triangle  add  up  to  180°, 
the  angles  of  a  quadrilateral  add  up  to  360°,  etc.  The  adjust- 
ment required  in  any  case  could  be  made  in  an  infinite  number 
of  ways,  but  the  adjustment  that  is  sought  is  the  one  that  assigns 
the  most  probable  values  to  the  various  angles  in  view  of  their 
actually  measured  values.  Since  all  the  angles  measured  are 
spherical  angles,  it  is  necessary  to  compute  the  spherical  excess 
in  work  of  any  magnitude  before  it  can  be  determined  to  what 
extent  the  measured  values  are  geometrically  inconsistent. 

If  all  the  triangulation  stations  (referred  to  mean  sea  level) 
were  connected  by  chords  instead  of  arcs,  we  would  have  a  net- 
work of  plane  triangles  perfectly  locating  all  the  stations,  and 
through  which  the  computations  could  be  carried  with  perfect 
accuracy,  provided  the  plane  angles  were  known  and  used. 
These  plane  angles  become  as  well  known  as  the  actually 
measured  spherical  angles  by  a  proper  reduction  for  spherical 
excess.  On  account  of  the  simplicity  and  saving  of  labor  the 
computations  in  practice  are  always  made  on  the  basis  of  plane 
triangles.  In  carrying  forward  the  azimuths  of  the  various  lines, 
however,  the  reduction  for  spherical  excess  must  be  restored  to 
the  adjusted  plane  angles,  and  a  further  allowance  made  for 
convergence  of  meridians,  as  explained  in  Chapter  V. 

56.  Spherical  Excess.  The  sum  of  the  angles  of  a  spherical 
triangle  is  always  greater  than  180°  by  an  amount  directly  pro- 
portional to  the  area  of  the  triangle  and  inversely  proportional 
to  the  surface  of  the  sphere,  the  value  of  the  increase  being  called 
the  spherical  excess.  It  follows  that  the  rule  must  also  hold  good 
for  any  spherical  polygon,  since  such  a  figure  can  always  be  divided 
up  into  spherical  triangles.  Owing  to  the  shape  of  the  earth, 
which  is  not  a  perfect  sphere,  the  spherical  excess  for  the  same 
area  decreases  slightly  as  we  advance  from  the  equator  toward 
the  poles;  except  for  very  large  areas  it  may  be  taken  as  1" 
for  every  76  square  miles,  the  true  value  for  this  area  being 
1".0035  +  in  latitude  18°  and  0".9925  +  in  latitude  72°.  It  may 
ordinarily  be  disregarded  entirely  where  the  area  is  less  than  10 


TRIANGULATION  ADJUSTMENTS  AND  COMPUTATIONS    89 

square   miles.     The   precise   formula   for  any   triangle   may   be 
written, 

_  area  X  (1  -  e2  sin2  <£)2 
~C~ 

in  which 

e  =  spherical  excess  in  seconds  of  arc; 
<p  =  latitude  at  center  of  triangle; 
loge2  =  7.8305026  -  10; 

f  1.8787228  (for  area  in  square  miles) 
log  C  =  \  9.3239906  (  "        square  feet) 

[  8.2920224  (  "        square  meters). 

For  logarithms  of  (1  —  e2  sin2  <£)  see  Table  IX. 

It  is  evident  that  neither  the  area  nor  the  latitude  need  be 
known  with  extreme  precision  for  the  present  purpose,  and  may 
be  estimated  before  any  adjustments  have  been  made. 

57.  Triangle  Adjustment.  The  failure  of  the  measured 
values  of  the  angles  of  a  triangle  to  add  up  to  180°  is  due  to  the 
spherical  excess  and  the  errors  of  measurement.  If  the  spherical 
excess  be  computed,  as  explained  in  the  previous  article,  the 
balance  of  the  discrepancy  represents  the  errors  of  measurement; 
or  in  order  words,  180°  -f  spherical  excess  —  sum  of  angles  = 
errors  of  measurement.  The  recognized  adjustment  for  spherical 
excess  is  a  deduction  of  one-third  of  the  total  excess  from  each 
angle,  which  is  not  mathematically  correct  unless  the  angles  are 
all  equal,  but  which  may  be  so  considered  in  any  case  that  arises 
in  practice ;  the  reason  for  this  is  found  in  the  fact  that  the  excess 
is  always  a  small  quantity  (rarely  reaching  60"),  and  also  that 
the  triangles  are  always  well  shaped  in  this  class  of  work.  The 
theoretical  adjustment  for  errors  of  measurement  is  to  divide  the 
amount  among  the  three  angles  inversely  as  their  weights;  if 
the  angles  are  of  equal  weight  this  results  in  correcting  each  angle 
by  one-third  of  the  error.  In  view  of  the  above  considerations 
the  failure  of  the  angles  of  a  triangle,  as  measured,  to  add  up  to 
180°  is  adjusted  as  follows: 

1.  If  all  the  angles  as  measured  are  considered  equally  reliable 
(of  equal  weight)  the  discrepancy  is  divided  equally  among  the 
three  angles.  The  spherical  excess  need  not  be  computed  in 
this  case,  unless  it  is  desired  for  other  purposes. 


90  GEODETIC  SURVEYING 

2.  In  important  work  where  the  angle  measurements  have 
different  weights,  each  angle  is  first  reduced  by  one-third  of  the 
spherical  excess,  and  then  corrected  for  the  errors  of  measure- 
ment inversely  as  its  weight. 

3.  In  small  triangles  or  work  of  minor  importance,  where  the 
angle  measurements  are  of  unequal  weight,  the  total  discrepancy 
is  divided  among  the  angles  inversely  as  their  weights. 

58.  The  Geodetic  Quadrilateral.  A  geodetic  quadrilateral  is 
formed  when  the  four  stations,  A,  B,  C,  D,  are  connected  as 
shown  in  Fig.  32.  The  size  of  the  largest  quadrilateral  occurring 


FIG.  32.— The  Geodetic  Quadrilateral. 

in  practice  is  relatively  so  small  as  compared  with  the  size  of  the 
earth  that  we  may  always  assume  without  material  error  that 
the  four  stations  lie  in  a  plane.  In  such  a  quadrilateral  one  side, 
as  AD,  must  be  known,  either  by  direct  measurement  or  connec- 
tion with  the  system;  and  the  eight  angles  a,  b,  c,  d,  e,  f,  g,  h, 
must  be  measured.  If  the  quadrilateral  is  of  sufficient  size  to 
require  it  the  measured  angles  must  be  reduced  for  the  spherical 
excess;  in  minor  work  this  may  be  distributed  equally  among 
the  eight  angles ;  in  more  important  work  each  of  the  four  triangles 
formed  by  the  intersection  of  the  diagonals  would  be  treated 
separately — that  is,  each  angle  would  be  reduced  by  one-third  of 
the  excess  appropriate  to  its  own  triangle.  In  the  plane  quadri- 
lateral ABCD  there  are  seven  angle  conditions  and  three  side 


TKIANGULATION  ADJUSTMENTS  AND  COMPUTATIONS    91 

conditions  that  must  be  satisfied  to  make  such  a  figure  geometric- 
ally possible,  and  these  ten  conditions  can  all  be  covered  by  three 
angle  equations  and  one  side  equation. 

The  seven  angle  conditions  are  as  follows  : 

1.  The  sum  of  the  eight  corner  angles  must  be  exactly  360°. 
This  furnishes  one  angle  condition. 

2.  The  opposite  angles  where  the  diagonals  cross  must  be 
equal.     This  furnishes  two  angle  conditions. 

3.  In  each  of  the  four  triangles  formed  among  the  stations, 
such  as  ABC,  the  sum  of  the  three  angles  must  be  exactly  180°. 
This  furnishes  four  angle  conditions. 

These  seven  conditions  are  so  involved,  however,  that  if  any 
three  independent  ones  are  satisfied  the  other  four  are  also  satis- 
fied. As  the  first  three  conditions  are  independent  all  the  angle 
conditions  will  be  satisfied  if  we  have 

a+b+c+d+e+f+g+h=  360°; 
a  +  b  =  e  +  /; 
c  +  d  =  g  +  h. 

The  three  side  conditions  arise  from  the  fact  that  each  unknown 
side  is  contained  in  two  different  triangles,  so  that  each  side  may 
be  found  by  two  independent  computations  which  must  give 
identical  results;  thus  the  unknown  side  BC  may  be  computed 
from  the  known  side  AD  through  the  triangles  ACD  and  BCD, 
or  through  the  triangles  ABD  and  ABC,  and  the  two  values  thus 
obtained  must  be  the  same.  These  three  conditions  are  not 
independent,  however,  for  if  any  one  of  them  is  satisfied  the  other 
two  are  also  satisfied.  It  is  well  to  note  that  all  the  seven  angle 
conditions  may  be  satisfied  without  satisfying  any  of  the  side 
conditions.  From  the  figure  we  have 


sm  d  sin  b  sm 

also 


sin  c 
whence 

BC    _  sin  a  sin  g  _  sin  /  sin  h 

AD       sin  b  sin  d       sin  c  sin  e  ' 
or 

sin  a  sin  c  sin  e  sin  g  _ 

sin  b  sin  d  sin  /  sin  h 


92  GEODETIC  SURVEYING 

which  is  called  the  side  equation.  When  this  equation  is  true 
the  side  conditions  will  all  be  satisfied.  Writing  the  side  equation 
in  logarithmic  form,  which  is  the  most  convenient  form  for  use, 
we  have 

(log  sin  a  +  log  sin  c  +  log  sin  e  +  log  sin  g) 

-  (log  sin  b  +  log  sin  d  +  log  sin/  +  log  sin  h)  =  0. 

59.  Approximate  Adjustment  of  a  Quadrilateral.  Assuming 
the  angles  to  have  been  measured  with  equal  care  (and  reduced 
for  spherical  excess,  if  necessary),  a  quadrilateral  of  moderate 
size  or  minor  importance  can  be  adjusted  with  sufficient  approx- 
imation and  with  comparatively  little  labor  by  the  method  here 
given. 

Referring  to  Art.  58,  the  equations  of  condition  which  must 
be  satisfied  are  as  follows : 

Angle  equations, 

a+b+c+d+e+f+g+h=  360°; 
a  +  b  =  e  +  /; 
c  +  d  =  g  +  h. 

Side  equation, 

(log  sin  a  +  log  sin  c  +  log  sin  e  +  log  sin  g) 

-  (log  sin  b  +  log  sin  d  +  log  sin  /  +  log  sin  ft)  =  0. 

The  adjustments  for  the  three  angle  equations  are  made  first; 
since  these  three  equations  are  independent  the  adjustments 
required  to  satisfy  them  may  be  made  in  any  order,  and  will  not 
disturb  each  other.  Since  the  angles  are  supposed  to  be  equally 
well  determined  the  adjustments  made  to  satisfy  any  one  of  the 
angle  equations  ought  to  have  the  same  value  for  each  angle 
affected.  Therefore,  if  the  eight  angles  fail  to  add  up  to  360°, 
each  angle  is  corrected  by  one-eighth  of  the  discrepancy;  thus 
if  the  sum  of  the  eight  angles  were  360°  00 '  08",  each  angle  would 
be  reduced  1".  If  a  +  b  fails  to  equal  e  +  /  each  angle  is  cor- 
rected by  one-fourth  the  discrepancy,  reducing  the  larger  side 
of  the  equation  and  increasing  the  smaller  one;  thus  if  a  -I  b 
exceed  e  +  f  by  8",  i  and  b  must  each  be  reduced  by  2"  and  e 


TRIANGULATION  ADJUSTMENTS  AND  COMPUTATIONS    93 

and  /  must  each  be  increased^by  2".  Similarly,  if  c  +  d  fails 
to  equal  g  +  h,  then  each  of  these  angles  must  be  corrected  for 
one-quarter  of  this  discrepancy. 

The  adjustment  for  the  side  equation  is  then  made  as  follows: 
Let  A,  B,  etc.,  represent  the  measured  angles  as  thus  far 

adjusted; 
I,  represent  the  value  of  the  first  member  of  the  side 

equation  when  A,  B,  etc.,  are  substituted  for  a,  b,  etc.; 
V,  represent  the  numerical  value  of  I; 
va,  vb ,  etc.,   represent  the  numerical   change  in  seconds 
required  in  A, B, etc.,  in  order  to  satisfy  the  side  equation; 
da  ,  db ,  etc.,  represent  the  tabular  differences  for  1"  for  log 
sin  A,  log  sin  B}  etc.     Then 

(log  sin  A  +  log  sin  C  +  log  sin  E  4-  log  sin  (?) 

-  (log  sin  B  +  log  sin  D  +  log  sin  F  +  log  sin  H)  =  I. 

Since  the  adjustment  of  the  angles  must  reduce  I  to  zero  (with  a 
minimum  change  in  each  angle),  it  is  seen  from  this  equation 
that  when  I  is  positive  the  first  four  terms  must  be  reduced  and 
the  last  four  increased,  and  vice  versa  when  I  is  negative.  This 
is  equivalent  to  saying  that  if  I  is  positive,  the  angles  A,  C,  E, 
and  G  must  be  reduced  if  less  than  90°,  and  increased  if  greater 
than  90°,  and  the  angles  B}  D,  F,  and  H  increased  if  less  than 
90°,  and  decreased  if  greater  than  90°;  and  that  if  I  is  negative, 
the  angles  A,  C,  E,  and  G  must  be  increased  if  less  than  90°,  and 
decreased  if  greater  than  90°,  and  the  angles  B,  D,  F,  and  H 
decreased  if  less  than  90°,  and  increased  if  greater  than  90°. 
It  therefore  only  remains  necessary  to  find  the  numerical  values 
of  the  corrections.  In  either  case,  in  order  that  I  may  vanish, 
the  numerical  sum  of  the  logarithmic  changes  must  equal  the 
numerical  value  of  I.  Since  changing  the  angle  A  by  va  changes 
log  sin  A  by  vada ,  etc.,  we  have 

vada  +  vcdc  +  vede  +  Vgdg  -f  vbdb  +  vddd  +  v/d,  +  vhdh  =  I', 

in  which  all  the  terms  are  to  be  made  positive.  Since  this  equation 
contains  eight  unknown  quantities,  va  ,  vc ,  etc.,  it  can  not  be  solved 
unless  some  additional  relationship  among  the  unknowns  is 
assumed.  This  relationship  is  found  in  the  fact  that  the  values 


94  GEODETIC  SURVEYING 

va,  vc,  etc.,  are  to  be  the  most  probable  ones;  and  it  is  generally 
admitted  that  the  most  probable  values  are  those  that  are  pro- 
portional to  their  influence  in  building  up  the  quantity  V  .  Thus 
if  da  is  twice  dc  ,  then,  second  by  second,  va  is  twice  as  effective 
as  vc  in  building  up  the  total,  I';  and  this  effectiveness  should  be 
recognized  by  allotting  twice  as  many  seconds  to  va  as  are  allotted 
to  vc  ,  and  so  on.  We  thus  have 

^a  :  vc  :  ve,  etc.  =  da  :  dc  :  de  ,  etc. 


But  if  -  =,   etc., 

Vc         dc  Ve         de' 

vada       da2  vcdc       dr2 

then  —  7-  =  ~TO  ,      —  7-  .=  -7-77  ,    etc., 

vcdc       dc2'  vede       de" 


or  vada  :  vcdc  :  vede  ,  etc.  =  da2  :  d2  :  de2,  etc. 

Referring  to  the  equation  to  be  solved,  therefore,  we  see  that 
V  is  to  be  divided  into  8  pieces  which  shall  be  in  the  ratio  of  the 
numbers  da2,  d2,  de2,  etc.,  giving  for  the  successive  terms  of  the 
equation  the  values 


d2l'      dfl' 

'    ' 


d2lf 
Va  °  =  lid2'     Vc  °  =       2'          ' 


Hence 


and  we  have  the  numerical  values 

/'   \ 

-1,     vc  =dc\,    etc., 


the  signs  of  these  corrections  having  been  determined  as  pre- 
viously explained. 

The  side-equation  adjustment  (having  been  derived  without 
regard  to  the  angle-equation  requirements)  will  probably  disturb 
the  angle-equation  adjustment  slightly,  but  seldom  seriously. 
If  necessary,  the  two  adjustments  may  be  repeated  in  turn  until 
both  are  satisfied  with  sufficient  approximation. 


TRIANGULATION  ADJUSTMENTS  AND  COMPUTATIONS    95 


o  g 
J3 

I? 


111! 


O"M 

*-  fl 


s 


3 


at 


05      |    05 


81    O 
°! 


s 


co 

v      00 


^  ! 


ol 


3 


3 


(N 


8 


CO      I    CO 

co    I  co 


§ 


H 


8 


8 


^ 


I     I 

KJ 


10  ^H  10 

Oi  »O  <N 

.T-i  1-1  (N 

II  II  II 


000 

XXX 

l-H     CO     (^ 


O5     O5     O5     O5 


<M     QJ.     O     ^H 

l®ii 

Isl* 

00     §5     05    t^ 
"35     ^     O5     O5 


»O    CO  CO    F-H 


°°.  I*    §5 
gj<>      II 


96 


GEODETIC  SURVEYING 


A  complete  example  of  adjustment  by  this  method  is 
worked  out  in  the  table  on  page  95.  In  this  particular 
case  the  side-equation  adjustment  has  disturbed  the  angle-equa- 
tion adjustment  to  a  maximum  extent  of  1".49.  If  this  approxi- 
mation is  not  as  close  as  desired  the  adjusted  values  may  be 
treated  like  original  values,  and  be  readjusted  by  the  same 
method.  A  second  adjustment  gives  the  following  values: 


a  =  46°  18'  38"  .47 

b  =  53  26  11  .92 

99  44  50  .39 

c  =42  11  27  .26 

d  =38  03  42  .35 

80  15  09  .61 

e  =  58  19  10  .54 

/  =  41  25  39  .90 

99  44  50  .44 

g  =  34  33  47  .38 

h  =  45  41  22  .18 

80  15  09  .56 

360  00  00  .00 


log  sin  =  9.8591959 
log  sin  = 

log  sin  =  9.8271126 

log  sin  = 

log  sin  =  9.9299248 
log  sin  = 

log  sin  =  9.7538238 
log  sin  = 


39.3700571 


9.9048230 


9.7899405 


9.8206448 


9.8546489 


39.3700572 


An  examination  of  these  values  shows  an  almost  perfect  adjust- 
ment. It  is  interesting  to  compare  the  results  of  both  the  first 
and  the  second  adjustment  with  the  results  of  the  rigorous  adjust- 
ment of  the  same  example  as  given  in  Art.  60. 

60.  Rigorous  Adjustment  of  a  Quadrilateral.  Assuming  the 
angles  to  have  been  measured  with  equal  care  (and  reduced  for 
spherical  excess,  if  necessary),  and  that  the  work  is  of  too  much 
importance  for  only  approximate  adjustment  (or  that  a  little 
extra  labor  on  the  computations  is  not  objectionable),  the  follow- 
ing method  may  be  used,  the  results  being  the  same  as  would  be 
obtained  by  the  method  of  least  squares. 

Referring  to  Art.  58,  the  equations  of  condition  to  be  satisfied 
are  as  follows: 


TEIANGULATION  ADJUSTMENTS  AND  COMPUTATIONS    97 

i 

Angle  equations,  & 

a  +  b  +  c  +  d  +  e+f  +  g+h=  360°; 
a  +  b  =  e  +  /; 
c  -f  d  =  g  +  h. 
Side  equation, 
(log  sin  a  +  log  sin  c  -f  log  sin  e  +  log  sin  g) 

—  (log  sin  6  +  log  sin  d  +  log  sin/  4-  log  sin  h)  =  0. 

As  in  the  case  of  the  approximate  method,  a  provisional  adjust- 
ment is  first  made  that  will  satisfy  the  angle  equations,  being 
made  in  the  same  way  as  there  explained  because  it  recognizes 
as  far  as  possible  the  fact  that  all  the  angles  have  been  measured 
with  equal  care.  This  adjustment  is  made  as  follows: 

If  a  +  b  +  c  +j  etc.,  fails  to  equal  360°,  correct  each  angle 
by  J  of  the  discrepancy. 

If  a  +  b  fails  to  equal  e  +  /,  increase  each  member  of  the 
smaller  sum  and  decrease  each  member  of  the  larger  sum  by  \ 
of  the  discrepancy. 

If  c  +  d  fails  to  equal  g  +  h,  increase  each  member  of  the 
smaller  sum  and  decrease  each  member  of  the  larger  sum  by  J 
of  the  discrepancy. 

The  side-equation  adjustment  is   then  made,  but  made  in 
such  a  way  as  will  not  disturb  the  angle-equation  adjustments. 
Let  A,  B,  etc.,  represent  the  angles  as  thus  far  adjusted; 

I,   represent  the  value  of  the  first  member  of  the  side 

equation  when  A,  B,  etc.,  are  substituted  for  a,  6,  etc.; 

va,  Vb,  etc.,  represent  the  total  corrections  in  seconds  to 

A,  B,  etc.,  to  satisfy  the  side  equation; 
x,  x\,  x2,  x3,  x4,  represent  the  partial  corrections  of  which 

va ,  vb,  etc.,  are  composed; 

da  ,  db ,  etc.,  represent  the  tabular  differences  for  1"  for  log 
sin  A,  log  sin  B,  etc.,  taken  as  positive  for  angles  less 
than  90°  and  negative  for  angles  greater  than  90°; 
then 

(log  sin  A  +  log  sin  C  +  log  sin  E  +  log  sin  G) 

—  (log  sin  B  +  log  sin  D  +  log  sin  F  +  log  sin  H)  =  Z; 


98  GEODETIC  SURVEYING 

and  in  order  that  the  logarithmic  corrections  shall  cause  I  to 
vanish  we  must  have 

(vafia  +  vcdc  +  vede  +  Vgdg)  —  (vbdb  +  vddd  +  vfdf  +  vhdh)   =  —  I, 

in  which  such  values  must  be  assigned  to  va,  vb ,  etc.,  as  will  not 
disturb  the  angle-equation  adjustments  already  made.  These 
adjustments  have  given  us 

(A  +  B)  +  (C  +  D)  +  (E  +  F)  +  (G  +  H)  =360°; 

(A  +B)  =(E  +  F); 
(C  +  D)  =  (G  +  H). 

It  is  evident  from  these  three  equations  of  condition  that  there 
are  only  two  possible  ways  in  which  the  adjusted  angles  A,  B, 
etc.,  can  be  modified  without  disturbing  the  angle-equation 
adjustments.  First,  any  correction  can  be  made  to  the  sum  of 
A  and  B,  provided  the  same  correction  is  made  to  the  sum  of 
E  and  F,  and  at  the  same  time  an  equal  and  opposite  correction 
is  made  to  each  of  the  other  two  sums;  since  the  two  angles 
of  any  sum  are  equally  reliable  the  same  numerical  change 
must  be  made  to  each  angle  and  will  be  denoted  by  x.  Second, 
any  group,  such  as  (A  +  B),  may  have  any  correction  applied  to 
one  of  its  members,  provided  an  equal  and  opposite  correction 
is  made  to  its  other  member;  these  corrections  are  independent 
of  the  first  correction  and  of  each  other,  and  will  be  represented 
by  x\t  x2,  £3,  and  x±.  In  accordance  with  the  above  considera- 
tions the  side-equation  adjustments  must  have  the  following 
relative  values: 

Va   =    +  X   +  Xi  Ve    =    +  X   +  X3 

Vb   =   +  X   —  Xi  Vf   =    +  X   —  X% 

vc  =    -  x  +  x2  Vg  •=  —  x  +  x± 

Vd   =    —  X   —  X2  V  h  =     —  X   —  X4 


TRIANGULATION  ADJUSTMENTS  AND  COMPUTATIONS    99 

i 

Substituting  these  values  in  oui*fconditional  side  equation 

(vada  +  vcdc  +  vede  +  Vgdg)  -  (vbdb  +  vddd  +  Vjdf  +  vhdh)  =  -  I, 
and  rearranging  the  terms,  we  have 

[(da  +  dd  +  de  +  dh)  -  (4  +  de+df  +  da)]x  +  (da  +  db)  X! 

+  (de  +  dd)x2  +  (de  +  d,)x3  +  (dg+  dh)x4  =  -I, 


which  for  convenience  we  write 

Cx  +  Cixi  +  C2x2  +  C3x3  +  C4x4  =  -  L 


Since  this  equation  contains  five  unknown  quantities  it  can  not 
be  solved  unless  some  additional  relationship  among  the  unknowns 
is  assumed.  The  most  probable  relationship  is  therefore  taken, 
namely,  that  the  unknowns  are  proportional  to  their  average 
effectiveness  per  angle  in  building  up  the  quantity  (  —  I)  .  Hence, 
since  x  affects  8  angles  and  the  other  unknowns  only  2  each,  we 
write 

C    C\    C2    C$    04      C     ~      ~      ~      ~ 
^  :  ^i  :  x2  :  o:3  :  x4  =  g  :  y  :  Y  :  Y  :  y  =  j-  :  Ci  :  C2  :  C3  :  C/4. 

But  if 


,-  =  = 

xi      Ci'     x2      C2'     x3      CB' 

then 

2 

2  C^_ 

2  '   ( 


or 

Cx  :  dxi  :  C2x2  :  C3x3  :  C4x±  =  ^  :  d2  :  C22  :  C32  :  C42. 

Referring  to  the  equation  to  be  solved,   therefore,  we  see 
that  (  —  Z)  is  to  be  divided  into  five  pieces  which  shall  be  in  the 


100  GEODETIC  SURVEYING 

C2 

ratio  of  the  numbers  — ,  Ci2,  C22,  C32,  C42,  giving  for  the  succes- 
sive terms  of  the  equation  the  values 

C2, 


C22  +  C32  +  C£  +  d2  +  C22  +  C32 


Hence,  writing  S  =  ^ —  — >  we  have 

-r  +  d2  +  C22  +  C32  +  C42 


C2  C 

Cx  =  —S,     whence      x  =  -T  S: 
4  4 


,  xl  =         ; 

C2x2  =  C22S,  "  x2  =  C2S; 

C3x3  =  C32S,  "  x3  =  C3S; 

C4x4  =  C42S,  "  x4  =  048. 

Combining  these  values  of  x}  x\,  x2,  etc.,  to  form  va  ,  vb  ,  etc.,  and 
applying  these  corrections  to  A,  B,  etc.,  we  obtain  the  most 
probable  values  of  the  angles  a,  b,  etc.,  consistent  with  the  geo- 
metrical necessities  of  the  figure  and  with  the  fact  that  all  the 
angles  were  measured  with  equal  care. 

A  complete  example  of  adjustment  by  this  method  is  worked 
out  in  the  table  on  page  101,  using  the  same  quadrilateral 
that  was  adjusted  by  the  approximate  method  (pages  95  and  96)  in 
order  to  compare  results.  It  will  be  noted  that  the  first  approxi- 
mate adjustment  has  a  maximum  variation  of  only  0".42  from 
the  rigorous  adjustment,  and  that  the  second  approximation 
comes  within  0".02  of  the  rigorous  values. 

61.  Weighted  Adjustments  and  Larger  Systems.  If  the 
measured  angles  of  a  triangle  have  different  weights,  the  adjust- 
ment is  made  as  already  explained.  If  the  measured  angles  of 
a  quadrilateral  or  other  figure  are  not  of  equal  weight,  the  adjust- 
ment is  best  made  by  the  method  of  least  squares. 


TEIANGULATION  ADJUSTMENTS  AND  COMPUTATIONS     101 


J-S 


m* 


for  Opp. 
Angles. 


58 


,     g 

7 


-§  i  t 


8 


I   g     U     I 


8 


3 


B 


S 

§    3 

J L_ 

!      I 
I  „  Is-, 

i 


§ 


2    8 


s 


g 


CO  GO  O  Tt*  »O 

•— '  t>-  lO  oo  »O 

d  rn  <M'  PH'  oi 

+  I  I  I  I 

II  II  II  II  II 

£•  t<»  r»»  r»  r* 

00  GO  GO  GO  00 

°£  °£  2  2  2} 

o  o  o  o  o 

o  d  d  d  d 

I  I  I  I  I 

X  X  X  X  X 


I  + 

II  11^ 

+  + 


0 


o  GO 


GO  CO    CO 


00    CO 


&  s  s  a 


*0 

I 
II 

£>U 


~*    05    O    CO    «0    O 


o 

d 


CO    01 

05    ^H 


102  GEODETIC  SUEVEYING 

In  work  of  moderate  extent  or  importance  a  system  composed 
of  a  series  of  triangles  or  quadrilaterals  would  have  each  triangle 
or  quadrilateral  independently  adjusted.  In  work  of  the  highest 
importance,  such  as  primary  triangulation,  the  entire  system 
would  be  adjusted  simultaneously  by  the  method  of  least  squares. 

62.  Computing  the  Lines  of  the  System.     After  a  figure  or 
system  is  satisfactorily  adjusted  the  distances  between  the  various 
stations  are  computed,  solving  each  triangle  in  order  (as  a  plane 
triangle)  by  the  ordinary  sine  ratio.     In  the  case  of  the  quadri- 
lateral the  two  diagonals  and  the  sides  adjacent  to  the  known 
side  (called  the  base)  are  computed  from  the  triangles  involving 
the  base;   the  side  opposite  the  base  is  then  computed  from  both 
the  triangles  in  which  it  occurs,  and  the  mean  of  the  two  results 
taken  as  its  value.    These  two  values  would  of  course  be  exactly 
alike  if  the   angle  adjustments  were  perfect,  but   these  adjust- 
ments are  only  correct  as  far  as  they  are  carried  out  decimally; 
a  material  disagreement  in  the  two  values  would  indicate  errors 
in  the  computations. 

63.  Accuracy  of  Triangulation  Work.     The  accuracy  of  this 
class  of  work  is  judged  by  measuring  a  check  base  at  the  end  of 
the  system,  if  the  work  is  of  moderate  extent,  with  intermediate 
check  bases  if  the  work  covers  a  large  territory.     The  length  of 
the  check  base  as  computed  through  the  triangulation  system 
should  agree  closely  with  its  measured  length.     In  triangulation 
work  by  the  U.  S.  Coast  and  Geodetic  Survey,  extending  over 
several  states  in  one  system,  extremely  close  results  are  reached. 
In  systems  600  to  800  miles  in  length  the  computed  and  measured 
values  of  check  bases  may  agree  within  fractions  of  an  inch. 


I 


CHAPTER  V 
COMPUTING  THE  GEODETIC  POSITIONS 

64.  The  Problem.  After  the  triangulation  system  has  been 
computed  as  described  in  the  last  chapter  the  relative  positions 
of  the  various  stations  are  known.  By  computing  the  geodetic 
positions  is  meant  computing  the  absolute  positions  (latitudes 
and  longitudes)  of  the  triangulation  stations  from  their  relative 
positions;  this  computation  can  be  made  if  we  have  the  latitude 
and  longitude  of  one  of  the  stations  and  the  azimuth  of  one  of 
the  lines  through  that  station,  provided  we  know  the  shape  and 
dimensions  of  the  earth.  The  problem,  then,  may  be  stated 
as  follows:  Given  the  latitude  and  longitude  of  a  station  and  the 
azimuth  and  distance  to  another  station,  to  find  the  latitude  and 
longitude  and  the  back  azimuth  at  the  second  station.  This 
problem  is  often  called  the  L.  M.  Z.  problem,  the  letters  meaning 
latitude,  longitude  (meridian),  and  azimuth.  The  back  azimuth 
at  the  second  station  will  seldom  be  the  same  as  the  forward 
azimuth  at  the  first  station,  on  account  of  the  convergence  of 
the  meridians.  Having  found  the  latitude,  longitude,  and  back 
azimuth  at  the  second  station,  the  azimuths  of  the  other  lines 
at  that  station  become  known  through  the  adjusted  angles  at 
that  station,  remembering  that  azimuths  are  counted  clockwise 
from  the  south  point  continuously  up  to  360°,  and  that  if  the 
spherical  excess  has  been  removed  from  any  angle  it  must  be 
restored  for  the  present  purpose.  By  proceeding  with  the  com- 
putations in  the  same  manner  from  station  to  station  we  obtain 
the  latitudes,  longitudes,  and  azimuths  for  the  whole  system. 
There  are  many  methods  of  solving  the  given  problem,  depending 
on  the  distance  involved  and  the  precision  required;  all  methods 
are  somewhat  complicated  on  account  of  the  shape  of  the  earth. 
Two  of  the  best  solutions  will  be  considered  after  discussing  the 
figure  of  the  earth. 

103 


104  GEODETIC  SUEVEYING 

65.  The  Figure  of  the  Earth.  It  is  doubtful  when  it  was  first 
realized  that  the  surface  of  the  earth  is  not  a  plane.  Early 
Greek  philosophers  believed  in  solid  figures  of  various  shapes. 
Aristotle  (340  B.C.)  gives  reasons  for  believing  the  shape  to  be 
spherical,  geometers  estimating  the  circumference  at  300,000 
stadia.  The  famous  School  of  Alexandria  appears  to  have  made 
the  first  actual  measurements  of  the  curvature  of  the  earth, 
and  hence  its  radius,  the  earliest  measurement  being  made  by 
Erastosthenes,  about  the  year  230  B.C.,  and  a  second  one  a  little 
later.  Erastosthenes  concluded  that  the  circumference  of  the 
earth  was  about  250,000  stadia  in  length,  but  the  exact  length  of 
the  stadium  is  now  unknown.  The  knowledge  which  the  Greeks 
obtained  of  the  size  and  shape  of  the  earth  was  lost  during  the 
declining  civilization  that  followed,  and  no  further  measurements 
were  made  for  upwards  of  a  thousand  years.  About  the  year  825 
the  Arabs  made  a  very  good  determination  of  the  radius  of  the 
earth  by  measuring  the  arc  of  a  meridian  on  the  plains  of  Mesopo- 
tamia. This  was  followed  by  another  lapse  of  about  700  years 
before  any  further  measurements  were  undertaken.  During 
the  middle  ages  Europeans  generally  believed  the  earth  to  be 
flat  until  about  the  15th  century,  when  a  few  men,  such  as  Colum- 
bus, declared  it  to  be  globular.  In  the  16th  century  general 
belief  in  the  spherical  shape  of  the  earth  was  again  established. 

From  the  earliest  measurements  to  the  present  time  the 
principle  employed  has  been  essentially  the  same,  but  a  very 
much  higher  degree  of  accuracy  is  now  reached  on  account  of  the 
great  refinement  in  detail.  The  fundamental  idea  is  to  obtain 
both  the  linear  and  the  angular  measure  of  the  arc  of  a  meridian, 
whence  the  distance  divided  by  the  number  of  degrees  gives  the 
length  of  one  degree,  and  this  multiplied  by  360  gives  the  length 
of  the  entire  circumference.  In  early  times  the  meridian  arc 
was  actually  staked  out  and  its  length  obtained  by  direct  meas- 
urement, but  modern  methods  of  measuring  and  computing  are 
so  improved  that  distances  measured  in  any  direction  may  be 
utilized.  The  angular  measure  of  the  arc  is  the  angle  between 
its  two  end  radii  (which  meet  near  the  center  of  the  earth),  and 
its  value  is  obtained  by  finding  the  latitude  at  each  end  and 
taking  their  difference. 

When  Newton  discovered  the  law  of  gravitation  late  in  the 
17th  century  he  proved  that  the  earth  as  a  revolving  plastic  body 


COMPUTING  THE  GEODETIC  POSITIONS  105 

subject  to  its  own  attraction  should  have  taken  the  form  of  a 
slightly  flattened  sphere,  while  an  arc  measured  in  France  between 
1683  and  1716  indicated  an  elongated  sphere.  To  settle  the 
question  an  arc  was  measured  in  the  equatorial  regions  of  Peru 
(1735-1741)  and  another  in  the  polar  regions  of  Lapland  (1736- 
1737),  which  showed  that  a  degree  of  latitude  was  longer  near 
the  pole  than  near  the  equator  and  that  Newton's  theory  was 
correct.  Since  these  dates  a  large  amount  of  geodetic  work  has 
been  done,  in  which  France,  Great  Britain,  Germany,  Russia, 
and  the  United  States  have  taken  a  leading  part.  Among  the 
more  recent  arcs  measured  may  be  mentioned  the  Anglo-French 
arc,  extending  from  the  northern  part  of  the  British  Isles  south- 
ward into  Africa;  the  great  Russian  arc,  extending  from  the 
Arctic  Ocean  to  the  northern  boundary  of  Turkey;  the  great  Indian 
arc,  extending  from  the  southern  point  of  India  to  the  Himalayas; 
the  European  arc  of  a  parallel,  extending  from  southern  Ireland 
eastward  to  central  Russia;  and  in  the  United  States,  the  trans- 
continental arc,  extending  along  the  39th  parallel  from  the 
Atlantic  Ocean  to  the  Pacific  Ocean,  and  the  eastern  oblique  arc, 
extending  parallel  to  the  Atlantic  coast  from  Maine  to  Louisiana. 
These  six  arcs  joined  end  to  end  would  reach  about  two-fifths 
of  the  way  around  the  earth. 

66.  The  Precise  Figure.  Various  names  have  been  applied 
to  the  earth  from  time  to  time  in  the  attempt  to  describe  its 
shape  more  exactly  as  our  knowledge  has  advanced.  Roughly 
it  may  be  called  a  sphere,  since  the  flattening  at  the  poles  is  rela- 
tively very  small;  a  model  with  an  equatorial  diameter  of  fifty 
feet  would  only  be  flattened  one  inch  at  each  pole.  As  the  result 
of  many  precise  measurements  the  shape  has  been  found  to  be 
such  that  with  considerable  exactness  any  section  parallel  to  the 
equator  is  a  circle,  and  any  section  through  the  poles  is  an  ellipse; 
the  figure  is  such  as  may  be  generated  by  revolving  an  ellipse 
about  its  minor  axis  and  is  called  an  oblate  spheroid.  To  be 
still  more  exact,  the  equatorial  section  is  not  exactly  circular 
but  very  slightly  elliptical,  so  that  a  section  in  any  direction 
through  the  center  would  be  an  ellipse;  such  a  figure  is  called 
an  ellipsoid.  Still  further  exactness  indicates  that  the  southern 
hemisphere  is  a  trifle  larger  than  the  northern,  and  that  all  polar 
sections  are  therefore  slightly  oval,  leading  to  the  name  ovaloid. 
As  a  matter  of  absolute  precision  no  geometrical  solid  exactly 


106  GEODETIC  SURVEYING 

represents  the  shape  of  the  earth,  and  this  has  been  recognized 
by  applying  the  special  name  geoid. 

67.  The  Practical  Figure.  All  the  computations  in  geodetic 
work  are  based  on  the  assumption  that  the  figure  of  the  earth 
is  an  oblate  spheroid;  this  is  found  to  be  amply  precise,  since  the 
variations  from  this  figure  are  relatively  very  small.  The  most 
important  determinations  of  the  elements  of  the  spheroid, 
founded  on  the  best  available  data,  are  those  made  by  Bessel  in 
1841  and  Clarke  in  1866.  Bessel' s  spheroid  is  still  largely  used 
in  Europe,  but  all  computations  in  the  United  States  are  made  on 
the  basis  of  Clarke's  spheroid,  which  conforms  better  to  the  actual 
surface  of  this  country.  According  to  Clarke's  comparison  of 
standards  a  meter  contained  3.2808693  feet,  a  result  which  is  now 
known  to  be  too  large.  In  the  legal  units  of  the  United  States 
the  meter  contains  exactly  39.37  inches,  which  equals  3.2808333 
feet,  a  value  which  is  believed  to  be  very  close  to  the  exact  truth. 
The  elements  of  Clarke's  spheroid  in  U.  S.  legal  units  are  as 
follows : 

f    6,378,276.5  meters,     log  =  6.8047033 
Semi-major  axis  =  a  =  (  ^^  feet,  log  -  7.3206875 

,       f    6,356,653.7  meters,     log  =  6.8032285 
Semi-minor  axis  -  b  = 


Ellipticity       = =  e  =       0.00339007,  log  =  7.5302093  -  10 

Eccentricity    =.Ja    ~      =  e  =  0.08227184,  log  =  8.9152513  -  10 
Eccentricity2  =  ^ ~  ^   =  e2  =  0.0067686580,  log  =  7.8305026  -  10 

7)          9QQ  QQ 

Ratio  of  axes  -  -  -  on^'no>  log  "  9-9985252  -  10 


68.  Geometrical  Considerations.  In  Fig.  33  the  ellipse 
WNES  represents  a  polar  section  of  the  earth,  in  which  WNES 
is  the  meridian;  NS,  the  polar  axis,  or  minor  axis  of  the  ellipse; 
WE,  the  equatorial  diameter,  or  major  axis  of  the  ellipse;  n, 
any  point  on  the  meridian;  nt,  the  tangent  at  n;  nlpm,  the  normal 
at  n,  or  the  direction  of  the  plumb  line  if  there  is  no  local  deflection ; 


COMPUTING  THE  GEODETIC  POSITIONS 


107 


np,  the  radius  of  curvature  dt  n;  no,  the  radius  of  the  small  circle 
or  parallel  of  latitude  at  n;  *f,  f,  the  foci  of  the  ellipse;  <£,  the 
latitude  of  the  point  n.  It  is  to  be  noted  that  the  normal  nm 
from  the  point  n  does  not  pass  through  the  center  c  (except  when 
n  is  at  the  poles  or  on  the  equator),  and  that  the  radius  of  curva- 
ture (and  hence  the  length  of  a  degree  of  latitude)  increases  from 
the  equator  to  the  poles;  that  the  radii  of  curvature  for  different 


FIG.  33. 

latitudes  on  a  meridian  do  not  intersect  unless  produced;  and 
that  for  different  latitudes  not  on  the  same  meridian  the  normals 
(which  include  the  radii  of  curvature)  do  not  intersect  at  all. 

Since  the  normals  for  two  points  of  different  latitudes  and 
longitudes  do  not  intersect,  they  do  not  lie  in  a  plane;  hence, 
Fig.  34,  page  108,  the  vertical  plane  at  A(AaB)  which  includes 
B  and  the  line  of  sight  from  A  to  B,  is  not  the  same  as  the  vertical 


108 


GEODETIC  SURVEYING 


{plane  at  B(BbA)  which  includes  A  and  the  line  of  sight  from  B  to  A. 
The  lines  which  these  planes  cut  at  the  surface  of  the  spheroid 
are  called  elliptic  arcs.  In  setting  points  from  A  to  B  an  observer 
at  A  would  mark  out  the  line  AaB,  while  an  observer  at  B  would 
mark  out  the  line  BbA;  the  greatest  discrepancy  between  the 
lines  would  be  practically  at  the  center,  and  under  extreme  con- 
ditions might  amount  to  about  an  inch  for  50  mile  lines  and  10 
feet  for  500  mile  lines;  the  angles  bAa  and  bBa  might  approx- 
imate 0".l  for  50  mile  lines  and  2".0  for  500  mile  lines.  For 
lines  100  miles  or  so  long,  therefore,  it  is  evident  that  the  two 
elliptic  arcs  may  usually  be  regarded  as  identical,  but  that  for 
greater  distances  the  question  may  often  be  of  considerable 


FIG.  34. 


importance.  If  an  observer  should  set  up  his  instrument  at  any 
intermediate  point  on  either  elliptic  arc  he  would  not  find  himself 
in  line  with  A  and  B;  if  he  sighted  on  A,  for  instance,  he  could  not 
sight  on  B  by  simply  transiting  his  telescope,  as  the  angle  between 
A  and  B  would  not  measure  180°.  An  alignment  curve  (as  repre- 
sented by  the  dotted  line  CD,  Fig.  34)  is  such  a  line  that  at  any 
intermediate  point  a  vertical  plane  can  be  established  that  will 
pass  through  both  end  stations;  as  seen  from  any  intermediate 
point  the  two  end  stations  are  always  180°  apart;  such  a  line  is 
a  line  of  double  curvature,  slightly  less  in  length  than  the  elliptic 
arcs  between  which  it  lies,  and  tangent  to  the  line  of  sight  at  each 


COMPUTING  THE  GEODETIC  POSITIONS  109 

end.  A  geodesic  line  is  the  shortest  line  that  can  be  drawn  between 
two  points  on  a  spheroid,  and  is  a  line  of  double  curvature  resem- 
bling the  alignment  curve,  but  the  reverse  curvature  is  not  so 
pronounced.  Between  any  two  points  on  the  earth  that  are 
actually  intervisible  all  the  lines  described  may  be  regarded  as  of 
equal  length. 

In  geodetic  work  the  term  latitude  always  refers  to  the  angle 
(j)  (Fig.  33,  page  107)  or  geodetic  latitude,  and  not  to  the  angle  ncd 
or  geocentric  latitude.  The  astronomical  latitude,  or  angular 
distance  from  the  equator  to  the  zenith,  is  the  same  as  the  geodetic 
latitude  except  where  there  is  local  deflection  of  the  plumb  line. 
By  longitude  is  meant  the  angular  distance  from  some  fixed  meridian 
(usually  Greenwich)  to  the  given  meridian,  positive  when  counted 
westward.  By  the  azimuth  of  a  line  (or  a  direction)  from  a  given 
point  is  meant  its  angular  divergence  from  the  meridian  at  that 
point,  counted  clockwise  from  the  south  continuously  up  to  360°. 
Thus  in  Fig.  34,  the  angle  DAa  is  the  azimuth  at  A  towards  B 
(AaB  being  the  line  of  sight  from  A),  and  the  angle  SBb  (clock- 
wise as  marked)  is  the  azimuth  from  B  towards  A .  The  azimuth 
(or  forward  azimuth)  of  a  line  means  taken  forward  along  the 
line,  and  back  azimuth  means  in  the  reverse  direction;  the 
azimuth  and  back  azimuth  at  the  same  point  differ  by  180°. 
The  angles  NAa  and  NBb,  inside  the  two  polar  triangles  NAB, 
are  called  azimuthal  angles,  the  angle  at  each  station  being  taken 
to  the  line  of  sight  from  that  station;  the  relation  of  these  angles 
to  the  azimuth  above  described  is  self  evident.  In  solving  either 
of  the  triangles  NAB  the  angles  at  both  A  and  B  must  be  taken 
in  the  same  triangle,  the  necessary  reduction  being  made  by 
means  of  the  auxiliary  angles  bBa  and  bA a. 

69.  Analytical  Considerations.  The  most  important  section 
of  the  spheroid  is  the  meridian  section,  Fig.  33,  page  107,  of  which 
N  and  R  are  the  principal  functions. 

Let  N  =  the  normal  nm; 

R  =  radius  of  curvature  np; 
r  =  radius  no  of  parallel  of  latitude ; 
T  =  tangent  nt; 
(j)  =  latitude  (geodetic) ; 
/?  =  geocentric  latitude; 
p  =  radius  vector  nc; 


110 


GEODETIC  SURVEYING 


then  from  analytical  geometry 


a(l-e2) 


_ 
(l-e2sin2</>)*' 

b2 
R  at  equator  =  —  , 

r  =  N  cos  (f>, 
nl  -  N(l  -  e2), 

b2 

tan  8  =  —  o  tan  d>, 
a2 


R  at  poles  =  -j-, 

T  =  N  cot  <£, 

d  =  N(\  -  e2)  sin  0, 

p  =  a(l  — e2sin2/3)*  (approx.), 


VRN  =  radius  of  osculating  sphere  at  n  =     _   2 
in  which  the  logarithms  of  the  constants  are  as  follows : 


-e2 


sn 


Quantity. 
a 

Metric. 

6.8047033 

Feet. 

7.3206875 

b 

6.8032285 

7.3192127 

e2 

7.8305026  -  10 

7.8305026- 

10 

(1  -  e2) 

9.9970504-10 

9.9970504- 

10 

a(l  -  e2) 

6.8017537 

7.3177379 

»v 

'1  -e2  =b 

6.8032285 

7.3192127 

b2 

=  a(\  -  e2) 

6.8017537 

7.3177379 

a 

a2 

a 

fi  SOfi17S1 

7  2991fi92 

9.9970504-10 


9.9970504-10 


The  section  of  next  importance  at  any  point,  after  the  merid- 
ian section,  is  that  which  is  cut  from  the  spheroid  by  the 
prime  vertical,  which  is  the  vertical  plane  at  the  given  point 
that  is  perpendicular  to  the  meridian  through  that  point.  The 
ellipse  that  is  thus  cut  from  the  spheroid  is  tangent  to  the 
parallel  of  latitude  through  the  given  point,  and  hence  a  straight 
line  run  east  or  west  from  any  point  is  commonly  called  a  tangent. 
The  radius  of  curvature,  RP,  of  a  prime-vertical  section  at  the 
point  where  it  originates  has  the  same  length  as  the  normal 
N  at  that  point,  that  is, 

a 


Rr,  =N  = 


(1  -  e2  sin2 


COMPUTING  THE  GEODETIC  POSITIONS 


111 


A  vertical  plane  at  a  given  point  that  is  neither  a  meridional 
plane  nor  a  prime-vertical  plane,  is  called  an  azimuth  plane;  such 
a  plane  cuts  an  azimuth  section  from  the  spheroid  and  traces 
an  azimuth  line  on  its  surface,  that  is,  a  straight  line  whose  initial 
direction  is  not  at  right  angles  to  the  meridian.  All  the  prop- 
erties of  an  azimuth  section  may  be  deduced  from  those  of  the 
prime-vertical  and  meridional 
sections.  Thus,  for  instance, 

Let    ft  =  azimuth  of  azimuth 

line  at  initial  point; 
N  =  normal  at  same  point  ; 
R  =  radius  of  curvature 

of  meridian  section 

at  same  point; 
Ra  =  radius  of    curvature 

of  azimuth  section 

at  same  point: 


then 


R 


70,  Convergence  of  the 
Meridians.  On  account  of  the 
convergence  of  the  meridians 
the  azimuth  of  a  line  varies 
from  point  to  point,  unless 
the  given  line  be  the  equator 
or  a  meridian.  By  the  con- 
vergence of  the  meridians  is 
meant  their  angular  drawing 
towards  each  other  in  passing  FIG  35 

from  the  equator  to  the  poles. 

Any  two  meridians  are  parallel  at  the  equator  or  have  a  zero 
convergence  (meaning  no  inclination  towards  each  other); 
in  moving  towards  the  poles  the  meridians  incline  more  and 
more  towards  each  other,  until  at  the  poles  the  convergence 
is  just  equal  to  the  difference  of  longitude.  Referring  to  Fig.  35, 
the  convergence  at  any  two  points,  n,  n',  which  are  in  the 
same  latitude  0i,  is  found  by  drawing  tangents  from  n  and  nf 


112  GEODETIC  SUEVEYING 

to  their  intersection  t  on  the  polar  axis,  in  which  case  the  angle 
6  is  the  convergence  for  those  two  meridians  for  the  common 
latitude  <j>i.  When  the  two  points  P  and  Pf  are  not  in  the  same 
latitude  the  convergence  for  the  middle  (average)  latitude  is 
understood;  so  that  if  <£  and  </>'  represent  the  latitudes  of  the 
two  points  we  may  write  in  any  case  </>i  =  i(</>  +  <£'),  and  n 
and  nr  represent  points  on  the  middle  parallel  of  latitude. 

Let  </>i  =  the  common  latitude  for  the  points  n  and  nr  (or  the 
average  latitude  for  any  two  latitudes  <f>  and  <£') ; 

A\  =  difference  of  longitude  for  the  two  meridians; 

no  =  r  =  radius  of  parallel  of  latitude  at  n; 

nt  =  T  =  tangent  at  n. 

From  the  figure 

Chord  nri  =  2T  sin  £0=  2  r  sin 
Substituting  r  =  T  sin  </>i, 

2T  sin  i0  =  2T  sin  <£i  sin 
or 

sin  \Q  =  sin  J(^)  sin  <£i, 

which  in  terms  of  the  latitudes  <t>  and  <£'  becomes 
sin  ±0  =  sin  i(JA)  sin  i(<£  +  <//). 

When  the  difference  of  longitude,  AX9  is  small,  0  will  also  be  small, 
and  we  may  write  with  great  closeness 

6  =  (M)  sin  *(*  +  <£'), 

in  which  6  will  be  in  the  same  unit  as  ^A  (usually  taken  in  minutes 
or  seconds).  Thus  in  an  average  latitude  of  40°  and  a  difference 
of  longitude  of  one  degree,  or  about  60  miles,  the  error  of  the 
approximation  would  be  less  than  the  one  thousandth  part  of 
a  second. 

Referring  to  Fig.  36,  let  rr'  be  a  straight  line  in  the  plane 
stv,  and  passing  as  close  as  possible  to  the  points  P  and  Pr.  In 
any  case  occurring  in  practice  the  angle  rpv  will  differ  but  very 
little  from  the  forward  azimuth  at  P  of  a  true  geodetic  line  from 
P  through  P',  and  the  angle  rp's  will  closely  represent  the  corre- 


COMPUTING  THE  GEODETIC  POSITIONS 


113 


spending  forward  azimuth  at*P'.     We  may  therefore  write  with 
great  closeness 

Change  of  azimuth  =  rp's  —  rpv. 
But  from  the  figure 

6  =  rpv  —  rp's, 
or 

Change  of  azimuth  =  —  6  =  —  (JA)  sin  i(<£  +  <£>'). 


Hence,  in  passing  from  one  station  to  another,  the  change  of 

azimuth  is  very  closely  the  same  in  numerical  value  as  the  corre- 

sponding convergence    of    the 

meridians.      The   error  in  the 

approximation   in    running   60 

miles  in  any   direction  in  the 

neighborhood    of    40°    latitude 

would  not  exceed  one  tenth  of 

a    second.      In    the    northern 

hemisphere   the   azimuth   of  a 

line  decreases  in  running  west- 

ward, and  increases  in  running 

eastward,    and   vice    versa    in 

the  southern  hemisphere.     The 

minus  signs  in  the  last  formula 

must  therefore  be  changed   to 

plus    in    the    southern    hemi- 

sphere.     In   running   approxi- 

mately east  and  west  in  about  FIGi  3(5. 

40°  latitude  the  change  of  azi- 

muth will  be  over  half  a  minute  per  mile.     The  back  azimuth 

of  a  line  is  equal  to  the  forward  azimuth  at  the  same   point 

plus  180°  (less  360°  if  this  number  is  exceeded). 

71.  The  Puissant  Solution.  Given  the  latitude  and  longitude 
of  a  station  and  the  azimuth  and  distance  to  a  second  station, 
the  problem  (Art.  64)  is  to  find  the  latitude,  longitude,  and 
back  azimuth  at  the  second  station.  The  Puissant  solution 
(as  modified  by  the  U.  S.  C.  &  G.  S.)  is  found  amply  precise 
if  the  distance  between  the  stations  does  not  exceed  about  1° 
of  arc  or  about  69  miles  (in  which  case  the  errors  of  the  com- 
puted values  might  run  from  0.001  to  0.003  seconds).  For  a 


114  GEODETIC  SURVEYING 

less  degree  of  accuracy  the  method  may  be  used  up  to  about 
100  miles.     The  Puissant  method  has  the  advantage  that  only 
seven  place  logarithms  are  required.     With  the  help  of  special 
tables  for  certain  factors  in  the  formulas  the  actual  work  of 
computation  is  not  very  great.     For  a  derivation  of  the  for- 
mulas, examples  of  their  use,  and  a  complete  set  of  tables,  see 
Appendix  No  .9,  Report  for  1894,  U  .S.  Coast  and  Geodetic  Survey. 
These  formulas  (in  slightly  different  form)  are  as  follows: 
Let         <£>  =  the  known  latitude  at  the  first  station; 
A  =  the  known  longitude  at  the  first  station; 
«  =  the  known  azimuth  at  the  first  station; 
4>'  =  the  unknown  latitude  at  the  second  station; 
A'  =  the  unknown  longitude  at  the  second  station; 
<x!  =  the  unknown  back  azimuth  at  the  second 

station; 

s  =  the  known  distance  between  the  stations; 
A,  B,  etc.,  =  certain  factors  required  in  the  formulas; 

then  by  successive  steps  we  have 

h  =  s  cos  OL  .  B, 
d</>  =     -  (h  +  s2  sin2  a.C  -  h.s2  sin2a-#), 

or  with  ample  precision 

d(f>  (for  15  miles  or  less)  =  —  (h  +  s2  sin2  a-C). 

In  either  case 


and 

<f>'  —  </)  +  J0  =  latitude  of  second  station; 

.A 

' 


COS 

and 

A'  =  A  +  4  X  =  longitude  of  second  station  ; 


or  with  ample  precision 

//a  (for  15  miles  or  less)  =  -  (^/)  sin  i(</>  + 


COMPUTING  THE  GEODETIC  POSITIONS  115 

which  agrees  with  the.result  of  Art.  70.  The  sign  of  ^«  is  for 
the  northern  hemisphere,  and  is  to  be  reversed  in  the  southern 
hemisphere.  Then 

a'  =  a  +  AOL  -f  ISO0  =  back  azimuth  at  second  station. 

In  the  above  formulas  the  values  of  J<£,  //A,  and  A  a  are  obtained 
in  seconds.  In  using  the  formulas  both  north  and  south  latitude 
are  to  be  taken  as  positive,  west  longitude  as  positive  and  east 
longitude  as  negative,  and  the  trigonometric  functions  are  to  be 
given  their  proper  signs.  The  lettered  factors  of  the  formulas 
have  the  following  values: 


A  =  A'  (I  -  *  sin*  D  = 


—  2 


e  sin 


B  =5'(l-e2sin2<£)°,  E  =  Ef(l  +  Stan2  <£)(!  -  e2  sin2  <£), 

C  =  C"(l  -  e2  sin2  <£)2  tan  <f>,  F  =  F'  (sin  0  cos2  0), 
G  =  value  determined  by  second  part  of  Table  II, 
in  which  the  logarithms  of  the  constants  are  as  follows: 


Constant. 
1'               l 

Metric.                            Feet. 
8^007918       1O        7  QQ^7*37fi       1O 

a  arc  1" 
1 

8.5126714  -  10      7.9966872  -  10 

1  40fiQ^S1        10        0  *374QfiQ7       in 

a(l-e2)  arcl" 
C'                    l 

2a2(l-e2)  arc  1 

/)'  =  |e2arcl" 

2.6921687  -  20       2.6921687  -  20 

E'=~  5.6124421  -  20      4.5804737  -  20 

6a2 

F'  =  Y>  arc2  l"  8.2919684  -  20       8.2919684  -  20 

With  the  help  of  these  constants  it  is  not  difficult  to  find 
the  values  of  the  factors  A  to  F  for  any  latitude.  If  the  distance 
s  is  given  in  meters  these  factors  may  be  taken  from  Table  II, 
at  the  end  of  the  book,  this  table  being  an  abridgment  of  the 


116  GEODETIC  SURVEYING 

Coast  Survey  tables  referred  to   (and  corrected  to  agree  with 
the  U.  S.  legal  meter  of  39.37  inches). 

72.  The  Clarke  Solution.  This  solution  of  the  problem  (Art. 
64)  is  adapted  to  greater  distances  than  the  previous  one,  being 
sufficiently  precise  for  the  longest  lines  (say  about  300  miles)  that 
could  ever  be  directly  observed.  It  has  the  advantage  of  being 
reasonably  convenient  in  use,  even  without  specially  prepared 
tables,  but  requires  not  less  than  nine  place  logarithms  for  close 
work,  on  account  of  the  size  of  the  numbers  involved.  In  this 
method  the  azimuthal  angles  are  used  in  the  computations 
instead  of  the  azimuths  themselves.  The  azimuthal  angles 
(shown  in  Fig.  34,  page  108,  and  explained  at  end  of  Art.  68), 
are  the  angles  (at  the  stations)  inside  the  polar  triangles  which 
are  formed  by  the  nearest  pole  and  the  two  stations,  the  relation 
to  the  corresponding  azimuths  being  always  self-evident.  The 
formulas  used  in  this  solution  (taken  from  Appendix  No.  9, 
Report  for  1885,  U.  S.  Coast  and  Geodetic  Survey,  but  modi- 
fied in  form)  are  as  follows: 

Let  (/>  =  the  known  latitude  at  the  first  station; 
A  =  the  known  longitude  at  the  first  station; 
a  =  the  known  azimuthal  angle  at  the  first  station  ; 
<j)f  =  the  unknown  latitude  at  the  second  station; 
),'  =  the  unknown  longitude  at  the  second  station; 
a'  =  the  unknown  azimuthal  angle  at  the  second  station; 

5  =  the  known  distance  between  the  stations; 

6  =  the  angle  between  terminal  normals; 

£,  =  auxiliary  azimuthal  angle  at  second  station; 
4  A  =  X'  —  X  =  difference  of  longitude  ; 
4(j)  ==  <£'  —  (j)  =  difference  of  latitude  ; 
Y  =  90°  —  (j>  —  co-latitude  at  first  station  ; 
N  =  normal  (to  minor  axis)  at  first  station  ; 
R  =  radius  of  curvature  of  meridian  at  middle  latitude; 
%(<f>  +  (£>')  =  middle  latitude. 


From  Art.  69, 


*-        a(1-e2) 


(1  —  e2  sin2  $)*'  [1  —  e2  sin2  £(<£  + 

Then 


COMPUTING  THE  GEODETIC  POSITIONS  117 

But  if  s  is  not  over  about  10Q,  miles  we  may  write  with  ample 
precision 

6  = 


AT  sin  I"' 

In  either  case  s  and  N  must  be  in  the  same  unit,  and  6  is  obtained 
in  seconds.  If  the  second  term  is  used  in  finding  0  the  approx- 
imate value  of  6  is  used  in  that  term.  The  value  of  this  second 
term  is  always  extremely  small.  Then 

c.- 


in  which  £  is  obtained  in  seconds  and  is  always  a  very  small 
quantity; 

D      sin  i(r—  6)      ,  a 
tan  P  =  -  —  f-f  -  ~  cot  -, 
)         2' 


~       cos|(r  —  0)         a 

tan  Q  =  -  ^-f  -  af  cot  ^ 

cosj  (7-  +  0)         2' 

from  which  values 

a'  =  P  +  Q  —  £  =  azimuihal  angle  at  second  station] 
Al=Q  -P-, 
^  =  X  +  AX  =  longitude  at  second  station. 

The  difference  of  latitude  is  found  from  the  formula 


in  which  ^  is  obtained  in  seconds,  and  in  which  s  and  72  must 
be  in  the  same  unit.  Then 

<f>'  =  (f>  +  A<j)  =  latitude  at  second  station. 

It  must  be  noted,  however,  that  the  4<t>  formula  requires  the 
use  of  R  for  the  middle  latitude,  which  is  not  known  .until  ^<£ 
is  found.  ^f</>  must  therefore  be  found  by  successive  approximation 
—  that  is,  an  approximate  value  of  R  must  first  be  used  to  obtain 
an  approximate  value  of  ^<£,  a  greatly  improved  value  of  R  thus 
becoming  available  to  find  a  much  closer  value  of  ^<£,  and  so  on. 


118 


GEODETIC  SURVEYING 


A  few  trials  will  soon  give  a  value  of  R  which  is  consistent  with  the 
value  of  $'  to  which  it  leads.  As  with  the  Puissant  formulas, 
both  north  and  south  latitude  are  to  be  taken  as  positive,  west 
longitude  as  positive  and  east  longitude  as  negative,  and  trig- 
onometric functions  used  with  their  proper  signs.  The  constants 
which  enter  into  the  above  formulas  have  the  following  values: 


Quantity. 

Log. 

a  (metric) 

6.8047033 

a  (feet) 

7.3206875 

e2 

7.8305026- 

10 

e2  sin2  I" 

6.4264506  - 

20 

6(1  -  e2) 

e2  sin  1" 

1   Q1AQA71  _ 

in 

4(l-e2) 


Quantity. 

a(l  —  e2)  (metric) 
a(l  -  e2)  (feet) 
(1  -  e2) 

sin  I" 

sin2  I" 
12 


Log. 

6.8017537 
7.3177379 
9.9970504  -  10 

4.6855749  -  10 
8.2919684-20 


When  the  distance  is  so  great  ^that  the  Clarke  solution  is  not 
satisfactory,  resort  must  be  had  to  more  direct  solutions,  requiring 
at  least  ten  place  logarithms.  The  solutions  by  Bessel  (1826) 
and  Helmert  (1880)  are  of  this  character. 

73.  The  Inverse  Problem.  In  this  case  the  latitude  and 
longitude  are  known  at  each  of  two  stations,  and  the  problem 
is  to  find  the  connecting  distance  and  the  mutual  azimuths. 
The  solution  may  be  effected  with  either  the  Puissant  or  the 
Clarke  formulas. 

By  the  Puissant  Formulas.  There  are  several  ways  of 
securing  the  desired  result;  the  one  here  given  is  chosen  on 
account  of  its  directness  and  simplicity.  By  transforming  and 
combining  the  formulas  in  Art.  71,  omitting  terms  which  are 
too  small  to  be  appreciable,  and  writing  x  and  y  for  the  resulting 
values,  we  have 


ssm  a  =  y= 


(J/l) 


cos  a  =  x  =  - 


from  which  we  obtain 


C-y2 


U  ,  </  •*' 

tan  a  =  —     and     s  =  -^ —  = . 

x  sin  a      cos  a. 


COMPUTING  THE  GEODETIC  POSITIONS 


119 


The  closest  value  of  s  is  obtained  from  the  fraction  whose  numer- 
ator is  the  smallest.    Then,  from  Art.  71, 


--  [ 


**--    (JX)  sin  i(*  +  f) 

A  a  (for  15  miles  or  less)  =  —(JX)  sin 

and  in  either  cast* 

a'  =  a  +  Aa  +  180°. 

Either  station  may  be  called  the  first  station,  so  that  the  problem 
may  be  worked  both  ways  as  a  check,  if  desired,  in  which  case 
Aa  need  not  be  computed  at  all.  As  in  Art.  71,  the  values 
^«,  ^(f>,  and  A\  are  expressed  in  seconds,  and  s  will  be  in  the 
same  unit  as  that  on  which  the  factors  A,  B,  etc.,  are  based. 

By  the  Clarke  Formulas.     In  this  method  the  desired  values 
are  found  by  successive  approximation,     The  Puissant  method 


FIG.  37. 

is  applied  first,  therefore,  to  obtain  as  close  an  approximation 
as  possible  to  begin  with.  The  approximate  values  of  s  and  a 
(changed  to  the  azimuthal  angle)  are  then  substituted  in  the 
Clarke  formulas,  calling  either  station  the  first  station,  and  Com- 
puting the  latitude  and  longitude  for  the  second  station.  The 
computed  values  will  usually  disagree  a  small  amount  with  the 
known  latitude  and  longitude  of  the  second  station,  and  a  new 
trial  has  to  be  made  with  s  and  a  slightly  changed,  and  so  on 
until  the  assumed  values  of  s  and  a  satisfy  the  known  con- 
ditions. The  disagreement  to  be  adjusted  is  always  very  small, 
and  when  all  the  circumstances  are  known  it  is  not  difficult  to 


120  GEODETIC  SURVEYING 

judge  which  way  and  how  much  to  modify  s  and  a  to  remove 
the  difficulty.  Referring  to  Fig.  37,  let  the  lines  NS  represent 
meridians,  the  line  CB  a  parallel  of  latitude,  and  A  and  B  the 
points  whose  latitude  and  longitude  are  known.  With  the  assumed 
distance  s  and  the  assumed  azimuthal  angle  «  suppose,  for 
instance,  that  the  computation  gives  us  the  point  B'  instead 
of  the  desired  point  B.  We  then  have 

BC  =  error  in  longitude  in  seconds  of  arc; 
B'C  =    error  in  latitude  in  seconds  of  arc; 


BB'  (in  seconds)  =  V  BC2  +  B'C2; 

b  =  BB'  in  distance  =  (BB')R  sin  1"  (approximately); 


BB'D  =  180°  -  a'  -  CB'B; 

b   cos   BB'D  =  B'D  =  approximate   error  in   the   assumed  value 
for  distance  s; 

b  sin  BB'D  . 

—  :  —  -jj—  =  BAD   (nearly)  in  seconds  =  approximate   error  in 

assumed  value  of  angle  a. 

74.  Locating  a  Parallel  of  Latitude.  For  marking  bound- 
aries, or  other  purposes,  it  often  becomes  desirable  to  stake  out 
a  parallel  of  latitude  directly  on  the  ground.  Points  on  the 
parallel  are  most  conveniently  found  by  offsets  from  a  tangent 
(Art.  69).  Thus  in  Fig.  38,  ABD  is  a  tangent  from  the  point 
A,  and  ACF  is  the  corresponding  parallel;  the  point  C  on  the 
parallel,  for  instance,  is  determined  by  the  offset  BC  and  the 
back-azimuth  angle  SB  A  .  It  is  seldom  desirable  to  run  a  tangent 
over  50  miles  on  account  of  the  long  offsets  required  ;  if  the  parallel 
is  of  greater  length  it  is  better  to  start  new  tangents  occasionally. 
The  computations  may  be  made  by  either  the  Puissant.  (Art. 
71),  or  the  Clarke  (Art.  72)  formulas,  which  are  much  simplified 
by  the  east  and  west  azimuths.  Using  the  Puissant  formulas, 
substituting  90°  (westward)  or  270°  (eastward)  for  a,  and  omitting 


COMPUTING  THE  GEODETIC  POSITIONS 


121 


inappreciable  terms,  we  have  with  great  precision  for  a  hundred 
miles  or  more 

A<f>  (in  seconds)  =  —  s2  •  C, 


whence 


.  ,  ,.  ,  N       f  running  W,  -f  1    s-A 

A  I  (in  seconds)  =  \          •      .» 

[  running  #,    —  J  cos  0' 


(in  linear  units)  =  —  (s2-C)  R  sin  I"  =  - 
N  N  N  N 


in  which  either  formula  may  be  used  as  preferred,  and  in  which 
all  linear  quantities  must  be  taken  in  the  same  unit.  The 
expressions  for  N  and  R  are  given  in  Art.  69.  For  the  change 
of  azimuth  we  have 

A     r  ^\      fN.  hemisphere,  —  1  r/  ...    .    ,  ,. 

A  a  (in  seconds)  =  ]  *  [  (A  X)  sin  J  (<£+<£')  +  (  J  ;)  3  •  F]  ; 

[b.  ,+  J 

or  for  the  field  work  (within  one-tenth  of  a  second), 

A     r  i  N       f  N.  hemisphere,  — 

Ja  (m  seconds)  = 


,  + 


sm 


It  is  seen  from  the  above  formulas  that  the  offsets  (in  seconds 
or  linear  units)  may  be  taken  to  vary  directly  as  the  square  of 
the  distance,  and  the  change  of  azimuth  directly  as  the  change 
of  longitude. 

In  actual  practice  the  point  A  may  have  to  be  located,  or 
may  be  given  by  description  or  monument;  in  either  case  the 
latitude  and  meridian  at  A  are  determined  by  astronomical 


122  GEODETIC  SURVEYING 

observations,  and  the  tangent  AB  (or  a  line  parallel  thereto) 
run  out  by  the  ordinary  method  of  double  centering.  At  the 
end  of  the  tangent  the  computed  value  of  the  back  azimuth 
should  be  compared  with  an  astronomical  determination;  in 
the  writer's  experience  on  the  Mexican  Boundary  Survey  with 
an  8-inch  repeating  instrument  (with  striding  level),  and  heliotrope 
sights  ranging  in  length  from  6  to  80  miles,  the  back-azimuth 
error  was  readily  kept  below  one-tenth  of  a  second  per  mile, 
regardless  of  the  number  of  prolongations  in  the  line.  The 
conditions  met  with  in  the  survey  referred  to  are  illustrated 
in  Fig.  39,  which  shows  also  the  adjustment  made  for  back- 
azimuth  error.  The  boundary  line  was  intended  to  be  the  parallel 
of  31°  47',  but  according  to  treaty  all  existing  monuments  had 
to  be  accepted  as  marking  the  true  line.  The  astronomical 
station  was  conveniently  located,  and  proved  to  be  slightly  south 
of  the  desired  parallel,  which  in  turn  passed  south  of  the  old 
monument  L.  When  the  last  point  on  the  tangent  was  reached 
the  back  azimuth  measured  less  than  the  theoretical  value, 
indicating  that  the  tangent  as  staked  out  swerved  slightly  to 
the  south  from  its  original  direction.  Assuming  all  corresponding 
distances  on  tangents  and  parallels  to  be  equal  and  the  azimuth 
error  to  accumulate  uniformly  from  A  to  d, 

Let  E  =  azimuth  error  at  d; 
Ei  =  azimuth  error  at  6; 
then 


DF  =  4<f>  (linear)  for  AD>}     BC  =  J0  (linear)  for  AB; 
dM  and  AL  are  known  by  measurement; 
FG  =  CH  =  AL; 
GM  =  dM  -  dD  -  DF  -  AL] 


~.. 
LG  Ad 

Hence  for  any  point  P,  on  the  adjusted  boundary,  we  have 
bP  =  bB  +  BC  +  CH  +  HP. 


COMPUTING  THE  GEODETIC  POSITIONS 


123 


CO 


or 

OF  CIVIL 


124  GEODETIC  SURVEYING 

75.  Deviation  of  the  Plumb  Line.  There  is  always  more  or 
less  uncertainty  at  any  station  as  to  the  plumb  line  hanging 
truly  vertical,  or  normal  to  the  surface  of  the  spheroid;  it  is 
not  uncommon  for  the  deviation  to  amount  to  10  or  more  seconds 
of  arc,  with  occasional  values  of  15  to  30  seconds.  This  fact 
is  forced  on  our  notice  in  a  number  of  ways;  if,  for  instance, 
the  computed  latitudes  and  longitudes  of  the  stations  in  a  triangu- 
lation  system  are  tested  by  astronomical  observations,  the  dis- 
crepancies are  often  greater  than  can  be  charged  to  either 
determination;  if  a  parallel  of  latitude  is  staked  out  and  tested 
astronomically  at  different  points,  the  same  discrepancies  appear. 
By  a  proper  combination  of  geodetic  and  astronomical  measure- 
ments involving  a  number  of  stations,  the  probable  deviation 
at  each  station  and  the  probable  errors  in  the  latitude  and  long- 
itude determinations  can  be  computed.  Astronomical  and 
computed  azimuths  disagree  for  the  same  reason,  and  require 
similar  adjustment.  In  moderate  sized  triangulation  systems, 
such  as  are  likely  to  engage  the  attention  of  the  civil  engineer, 
adjustments  of  this  kind  are  rarely  called  for;  but  in  extended 
systems  astronomical  latitudes,  longitudes,  and  azimuths  are 
taken  at  many  stations,  in  order  that  such  adjustments  may  be 
made. 


m- 

CHAPTER  VI 
GEODETIC    LEVELING 

76.  Principles  and  Methods.     Leveling  is  the  operation  of 
determining  the  relative   elevations  of   different  points  on  the 
surface  of  the  earth.     By  relative  elevation  is  meant  the  difference 
of  elevation  between  any  two  points  compared.     The  absolute 
elevation  of  a  point  is  its  elevation  above  some  particular  point 
or  surface  of  reference,  mean  low  water,  for  instance;  in  geodetic 
work  elevations  are  commonly  referred  to  mean  sea  level.     A 
level  line  is  a  line  having  the  same  absolute  elevation  at  every  point. 
By  geodetic  leveling  is  meant  that  class  of  leveling  in  which  extra 
precision  is  sought  by  refinement  of  instruments  and  methods. 

Three  principal  methods  are  available  for  determining  dif- 
ferences of  elevation,  (A)  Barometric  Leveling,  (B)  Trigonometric 
Leveling,  (C)  Precise  Spirit  Leveling.  Barometric  leveling,  based  on 
determinations  of  atmospheric  pressure,  is  briefly  treated  below 
on  account  of  its  usefulness  in  reconnaissance  work.  Geodetic 
leveling  is  generally  understood  to  mean  either  trigonometric 
leveling,  based  on  vertical  angles  (corrected  for  curvature  and 
refraction),  or  precise  spirit  leveling,  which  differs  from  ordinary 
spirit  leveling  only  in  the  refinement  of  its  details. 

77,  Determination  of  Mean  Sea  Level.    By  mean  sea  level  is 
meant  the  average  elevation  of  the  surface  of  the  sea  due  to 
its  continual  change  of  level;    and  not,  as  might  be  supposed, 
the  mean  elevation  of  its  high  and  low  waters.     In  order  to 
average  out  the  irregularities  due  to  winds  and  other  causes 
the  observations  at  any  point  should  extend  over  a  period  of 
several  years.     Further,,  since  tidal  variations  are  relatively  large 
during  a  lunar  month,  only  complete  lunations  can  be  allowed 
in  the  reductions;    if  any  storm  period,  for  instance,  is  rejected 
on  account  of  its  excessive  irregularities,  that  entire  lunation 
must  be  rejected. 

Observations  of  the  varying  elevation  of  the  surface  of  the 
sea  are  best  made  by  means  of  automatic  tide  gauges.  An 

125 


126  GEODETIC  SURVEYING 

automatic  or '  self-registering  tide  gauge  consists  essentially  of 
a  well  made  clock  and  attached  mechanism,  by  which  a  sheet 
of  paper  is  drawn  continuously  past  a  pencil  point  which  is  moved 
crosswise  of  the  paper  by  connection  with  a  float;  a  rising  and 
falling  curve  is  thus  traced  on  the  paper,  in  which  the  ordinate 
of  any  point  shows  the  elevation  of  the  water  at  the  time  indi- 
cated by  the  corresponding  abscissa.  The  float  moves  up  and 
down  in  a  vertical  box  admitting  water  only  through  a  small 
opening  in  the  bottom,  which  practically  prevents  oscillation 
of  the  float  by  wave  action.  A  catgut  cord  or  fine  wire  connects 
the  float  with  the  pencil  through  a  suitable  reducing  mechanism. 
Pin  points  are  often  arranged  to  prick  the  even  hours  on  the 
paper.  The  clock  is  often  designed  to  run  a  week  without 
rewinding,  and  the  paper  to  last  a  month  without  changing. 
A  scale  of  one  inch  per  foot  and  J  of  an  inch  per  hour  makes 
a  very  good  record. 

A  staff  tide  gauge  is  always  placed  as  near  as  possible  to  the 
automatic  gauge,  and  its  zero  point  connected  by  accurate 
leveling  with  a  permanent  bench  mark  near  by.  At  least  once 
a  week  the  attendant  carefully  raises  and  lowers  the  float  so  that 
the  pencil  of  the  automatic  gauge  will  mark  the  true  direction 
of  the  ordinates  at  that  time;  and  near  the  ordinate  thus  made 
he  records  the  date,  the  staff  reading,  and  the  clock  reading  and 
error.  The  attendant's  visits  should  be  so  timed  that  his  staff 
readings  will  be  alternately  near  high  and  low  water,  thus  fur- 
nishing scales  for  different  parts  of  the  sheet  that  will  practically 
neutralize  errors  due  to  stretching  or  shrinking  of  the  paper  or 
float  connections.  Hourly  ordinates  are  drawn  on  all  the  records 
obtained  at  a  station,  and  the  average  value  of  these  ordinates 
is  taken  as  the  staff  reading  of  mean  sea  level  at  that  station. 
The  relation  of  the  permanent  bench  mark  to  the  zero  of  the 
staff  having  been  determined,  as  previously  described,  the  ele- 
vation of  the  bench  mark  with  reference  to  mean  sea  level  becomes 
known,  and  furnishes  the  basis  of  the  precise  level  lines  that 
are  extended  to  inland  points. 

A.   BAROMETRIC  LEVELING 

78.  Instruments   and   Methods.      The  instruments  available 
are  the  familiar   types   of    aneroid   and   mercurial   barometers. 


GEODETIC  LEVELING  127 

The  mercurial  barometer  is  the  ^tandard  instrument  for  indicating 
atmospheric  pressure,*  but  lacks  the  aneroid's  advantage  of 
convenience  in  portability.  The  aneroid  barometer  is  decidedly 
inferior  to  the  mercurial  barometer  as  a  pressure  indicator, 
but  is  sufficiently  accurate  for  many  purposes,  such  as  recon- 
naissance work.  Pocket  aneroids  (about  3  inches  in  diameter) 
are  found  to  be  as  reliable  as  the  larger  sizes.  Aneroids  are 
intended  to  read  the  same  as  mercurial  barometers  under  the 
same  conditions,  being  compensated  for  the  effect  of  tempera- 
ture on  their  own  construction;  they  are  not  compensated  for 
the  effect  of  temperature  on  atmospheric  pressures.  The  aneroid 
requires  careful  handling,  should  be  kept  in  its  case  at  all  times 
and  away  from  the  heat  of  the  body,  should  be  read  in  the  open 
air  and  in  a  horizontal  position,  and  should  be  gently  tapped 
when  reading  to  overcome  any  friction  among  its  moving  parts. 

If  all  the  conditions  were  the  same  at  two  different  stations, 
the  difference  in  atmospheric  pressure  would  correspond  to  the 
difference  in  altitude;  for  points  not  over  about  100  miles  apart 
the  conditions  may  be  assumed  to  be  nearly  the  same  at  the  same 
time  in  ordinary  calm  weather.  Two  barometers  are  necessary 
for  good  work,  the  office  barometer  which  is  kept  at  the  reference 
station,  and  the  field  barometer,  which  is  carried  from  point 
to  point.  If  the  office  barometer  is  an  aneroid  it  must  be  stand- 
ardized, that  is,  adjusted  by  the  small  screw  at  the  back  until  it 
reads  the  same  as  a  mercurial  barometer.  During  the  period  of  ob- 
servations the  office  barometer  and  attached  thermometer  are  read 
at  regular  intervals  (about  15  or  30  minutes),  so  that  by  inter- 
polation the  readings  are  assumed  to  be  known  for  any  instant. 
The  time  and  temperature  are  recorded  whenever  a  field  reading 
is  taken,  so  that  comparison  may  be  made  with  the  office 
readings  for  the  same  time.  If  the  field  barometer  is  an  aneroid 
its  readings  will  need  correction  for  initial  error  and  inertia. 
Before  starting  out  to  take  readings  with  the  field  barometer 
it  is  compared  with  the  office  barometer  and  any  difference  is 
its  initial  error,  which  will  affect  all  its  readings  to  the  same 
extent.  On  returning  to  the  office  after  one  or  more  observations 
the  field  barometer  is  again  compared  with  the  office  one,  and 
the  amount  by  which  the  initial  error  has  changed  is  called 
the  inertia  error;  this  error  is  distributed  among  the  different 
readings  in  proportion  to  the  elapsed  time. 


128 


GEODETIC  SURVEYING 


79.  The  Computations.  The  complete  barometric  formula 
(for  which  see  Appendix  No.  10,  Report  for  1881,  U.  S.  Coast 
and  Geodetic  Survey)  is  very  complicated  and  the  smaller  terms 
are  generally  omitted  in  ordinary  work.  Assuming  all  readings 
reduced  to  the  standard  of  the  office  barometer, 

Let  H  =  elevation  in  feet  of  the  office  barometer  above  a 
plane  corresponding  to  a  barometric  pressure  of 
30  inches  for  dry  air  at  a  temperature  of  50°  F. ; 

h  =  the  same  for  the  field  barometer; 

B  =  reading  of  office  barometer  in  inches; 

b  =  corrected  reading  of  field  barometer  in  inches; 

t  =  Fahrenheit  temperature  at  office  barometer; 

t'  —  Fahrenheit  temperature  at  field  barometer; 

/  i // 
C=  correction  coefficient  for  mean  temperature  — ^—  for 

average  conditions  of  humidity; 
z=  difference  of  elevation  of  the  two  barometers  in  feet; 

then  we  have,  nearly, 


and 


Of)  Of) 

H  =  62737  log  22         h  =  62737  log  2" 
±>  b 


C), 


in  which  H  and  h  may  be  obtained  from  Table  III,  and  C  from 
Table  IV  opposite  (t  +  t'). 

Example.     In  the  following  table  the  field  observations  were  taken  with 
an  aneroid  and  require  the  corrections  described  above. 

FIELD  NOTES  AND  REDUCTIONS,  MAY  17,  1910. 


Station. 

Time. 

Barom. 

Temp. 

Initial 
Corr. 

Inertia 
Corr. 

Thermom. 
Corr. 

A 

8.00A.M. 

29.124 

73°  F. 

+  0.040 

+  1° 

B 

11.10A.M. 

28.247 

70°  F. 

+  0.040 

-0.006 

+  1° 

C 

1.30P.M. 

29.216 

79°  F. 

+  0.040 

-0.011 

+  1° 

A 

4.00PM. 

29.182 

79°  F. 

+  0.040 

-0.016 

+  1° 

GEODETIC  LEVELING  129 

OFFICE  NOTES  AND  RsbucTioNS,  MAY  17,  1910. 


Station, 

Office. 

Field  (reduced). 

Diff.  Elev. 

Elevation. 

Barom. 

Temp. 

Barom. 

Temp. 

A 

29.164 

74°  F. 

29.164 

74°  F. 

1867  ft. 

B 

29.179 

76°  F. 

28.281 

71°  F. 

897 

2764  ft. 

C 

29.189 

79°  F. 

29.245 

80°  F. 

-55 

1812  ft. 

A 

29.206 

80°  F. 

29.206 

80°  F. 

1867  ft. 

From  Table  III  (by  interpolation) 


6  =  28.281, 
B  =  29.179, 


h  =  1607 
H  =     756 

-  H  =     851 


6  =  29.245, 
B  =  29.189, 


H  = 


694 
746 


h-  H  =  -52 


From  Table  IV  (by  interpolation) 

147°,  C  =  +  0.0544 
851  X  0.0544  =  46  + 
851  +  46  =  897 


159°,  C  =  +  0.0667 

-  52  X  0.0667  =  -  3  + 

-  52  +  (  -  3)  =  -  55 


In    this  example   the   elevation  of    station  A    was   known   from   previous 
determinations . 

80.  Accuracy  of  Barometric  Work.  For  exploration,  recon- 
naissance, and  other  classes  of  work  where  close  results  are  not 
required,  the  barometer  serves  a  very  useful  purpose.  For 
stations  only  a  few  miles  apart,  or  not  differing  much  in  altitude, 
the  errors  in  the  determinations  may  not  exceed  a  few  feet. 
For  long  distances  or  large  differences  in  altitude  the  results 
are  very  disappointing,  notwithstanding  that  the  utmost  refine- 
ments of  theory  and  practice  are  employed,  and  daily  readings 
averaged  for  a  number  of  years.  In  general  the  values  obtained 
in  the  heat  of  the  day  are  too  great,  and  in  the  morning  and  evening 
too  small;  and  similarly  too  great  in  summer  and  too  small  in 
winter.  Professor  Whitney,  in  his  Barometric  Hypsometry,  gives 
the  results  of  three  years'  observations  at  Sacramento  and  Summit, 
California,  from  October,  1870,  to  October,  1873,  in  which  the 
monthly  average  determinations  of  the  difference  of  elevation 


130  GEODETIC  SURVEYING 

varied  from  6900  to  7021  feet,  the  average  for  the  three  years 
being  6965  feet;  according  to  railroad  levelings  the  true  dif- 
ference is  6989  feet.  Summit  is  about  77  miles  from  Sacramento 
in  an  air  line;  the  altitude  of  Sacramento  is  about  30  feet  above 
mean  sea  level.  The  example  given  is  a  fair  illustration  of 
the  general  experience  in  this  class  of  work,  with  plus  and  minus 
errors  about  equal.  The  chief  source  of  error  in  barometric 
work  seems  to  be  due  to  the  lack  of  knowledge  of  the  true 
average  temperature  of  the  air  column  between  the  levels  of 
any  two  given  stations,  the  mean  of  the  station  temperatures  be- 
ing only  a  fair  approximation. 

B.     TRIGONOMETRIC  LEVELING 

81.  Instruments  and  Methods.  Trigonometric  leveling  can 
be  done  with  any  instrument  capable  of  measuring  angles 
of  elevation  and  depression,  but  good  work  can  be  done 
only  when  the  angles  can  be  measured  with  precision.  While 
the  ordinary  surveyor's  transit  may  read  vertical  angles  only 
to  the  nearest  minute,  a  fine  altazimuth  instrument  may  be 
provided  with  micrometer  microscopes  reading  such  angles  to 
single  seconds.  In  round  numbers  a  minute  of  arc  corresponds 
to  a  foot  and  a  half  per  mile,  and  a  second  to  three-tenths  of  an 
inch;  with  moderate  sized  vertical  angles,  such  as  would  usually 
occur  in  trigonometric  leveling,  the  resulting  effect  in  altitude 
is  practically  the  same.  It  is  presumed  that  the  observer  under- 
stands how  to  adjust  and  use  his  particular  instrument  to  the 
best  advantage. 

The  elevation  of  a  station  from  which  the  open  sea  is  visible 
can  be  determined  by  measuring  the  angle  of  depression  to  the 
sea  horizon.  The  difference  of  elevation  of  two  stations  whose 
distance  apart  is  known  can  be  determined  by  measuring  the 
angular  elevation  of  one  of  them  as  seen  from  the  other,  con- 
stituting an  "observation  at  one  station,"  or  by  measuring  the 
angular  elevation  of  each  station  as  seen  from  the  other,  con- 
stituting "reciprocal  observations."  From  the  nature  of  the 
case  the  effects  of  curvature  and  refraction  are  necessarily  involved 
in  any  form  of  trigonometric  leveling.  The  best  results  are 
obtained  between  9.00  A.M.  and  3.30  P.M.,  during  which  time  the 
refraction  has  its  least  value  and  is  comparatively  stationary. 


GEODETIC  LEVELING 


131 


82.  By  the  Sea  Hprizon  Method.     If  a  station  is  so  situated 
as  to  command  a  view  of  the  open  sea  its  elevation  above  the 
surface  of  the  water  may  be  determined  by  measuring  the  angle 
of  depression  to  the  sea  horizon.     The  advantage  of  this  method 
lies  in  the  fact  that  no  distance  is  required  to  be  known.    Fig.  40 
represents   the  vertical  plane  of  the  measured  angle,  in  which 
A   is   the  station  whose  elevation  is 
desired;   SS  is   an  elliptic   arc  at  the       F 
level  of  the  sea  horizon,  but  it  is  here 
assumed  to    be  the    arc   of   a   circle; 
AE  is  a  straight  line  from  A  tangent 
to  the  arc  SS  at  the  point  E  or  true 
sea  horizon;   BC  (on  the  vertical  line 
AC)  is  the  radius  of  the  arc  SS,  and 
is  assumed  to  be  equal  to  the  mean- 
sea-level  radius  of  the  section  for  the 
point  A,  the  point  C  being  in  general 
not  at  the  center   of  the   earth.     E' 
is  the   false   horizon   caused   by  the 
refraction  of  light;    d  is  the  apparent 
and  C  the   true   angle    of  depression 
to    the    sea   horizon;    and    BD    is    a 
tangent    at    B.      From    well    known 
geometrical     principles     the     angles 

GAD,  ADB,  and  BCE  are   equal,  and  the    line  DC  bisects    the 
angle  at  C. 

Let  R  =  BC  =  the  mean-sea-level   radius   of  the  section  for 

the  point  A ; 

C  =  the  true  angle  of  depression  =  angle  at  center; 
d  =  the  apparent  angle  of  depression; 
Z  =  90°  +  d  =  apparent  zenith  distance  of  sea  horizon; 
m  =  coefficient  of  refraction; 
h  =  AB  =  elevation  of  station  A  above  surface  of  sea; 

then  from  the  figure  we  have 

h  =  BD  tan  C, 

C 
BD  =  #tan  - 


h  =  R  tan  —  tan  C; 


132  GEODETIC  SURVEYING 

or  since  C  is  always  a  small  angle  (rarely  60'), 

h  = 


On  account  of  refraction  (Art,  14)  the  observer  does  not  sight 
along  the  true  line  AE,  but  in  the  direction  of  the  dotted  line 
from  A,  which  is  tangent  to  a  curved  line  of  sight  from  A  to 
the  false  horizon  Er.  The  practical  result  of  the  refraction  is 
to  make  the  measured  angle  6  too  small  by  the  amount  mC, 
so  that 

d  =  C-mC, 


l-m; 
and 

H(dy2tan21"- 

whence  by  transposition 

h  =  , 


in  which  h  and  r  must  be  taken  in  the  same  unit,  and  3  must 
be  taken  in  seconds.  By  many  experiments  the  mean  value  of 
m  on  the  New  England  coast  has  been  found  to  be  0.078  •  if  we 
use  this  value  we  may  write 


=  9.1406579-  20. 


(tnn2  1"    \ 
2^^)  =  9. 


In  order  to  secure  the  best  results  it  is  necessary  to  measure 
the  azimuth  of  the  plane  in  which  the  angle  of  depression  is 
taken,  and  use  the  mean-sea-level  value  of  R  for  this  azimuth 
and  the  latitude  of  the  station.  This  value  may  be  taken  from 
Tables  V  and  VI,  or  computed  as  explained  in  Art.  69.  If  errors 
which  may  range  up  to  say  about  1  in  300  are  not  objectionable, 
we  may  use  a  mean  value  of  R  and  write 


tan2 


f  metric,  5.9446244-  10  1 
1  feet,       6.4606086  -  10  j 


GEODETIC  LEVELING  133 

'  •-  '* 

f  metric.  0.000088  a2]    , 
*-(feet,       0.000289  S3  j(appr°xlmate>' 

in  which  d  must  be  taken  in  seconds  of  arc. 

83.  By  an  Observation  at  One  Station.  When  the  distance 
between  two  stations  is  known  their  difference  of  elevation  can 
be  computed  if  the  vertical  angle  of  either  as  seen  from  the 
other  is  measured.  The  advantage  of  this  method  over  the 
reciprocal  method  (Art.  84)  lies  in  the  economies  due  to  occupying 
only  one  station,  but  the  results  are  not  likely  to  be  so  good 
on  account  of  the  uncertainty  in  the  assumed  value  for  the 


coefficient  of  refraction.  Fig.  41  represents  a  plane  through 
the  two  stations  A  and  B,  taken  vertical  at  their  middle  lat- 
itude, and  assumed  to  be  vertical  at  both  stations;  SS  is  the 
elliptic  arc  cut  from  the  spheroid,  but  it  is  here  assumed  to  be  the 
arc  of  a  circle;  the  radius  of  the  arc  SS  is  taken  as  the  mean- 
sea-level  radius  of  the  section  at  the  middle  latitude,  the  center 
C  being  in  general  not  at  the  center  of  the  earth;  AC  and  BC 
are  drawn  to  the  center  C  and  assumed  to  be  vertical;  Z  is  the 
apparent  zenith  distance  of  A  as  seen  from  B,  and  is  in  error 
by  the  small  angle  mC  due  to  refraction. 


134  GEODETIC  SURVEYING 

Let  h  =  AM  =  elevation  of  station  A  above  mean  sea  level; 
h'  =  BN  =  elevation  of  station  B  above  mean  sea  level; 
K  =  MN  =  mean-sea-level  distance  between  stations 

A  and  £; 
R  =  MC  =  mean-sea-level  radius  of  section  at  middle 

latitude  between  A  and  B' 
C  =  central  angle  A  CB; 

Z  =••  apparent  zenith  distance  of  A  as  seen  from  B'} 
a  =  90°  —  Z  =  apparent  elevation  of  A  as  seen  from  B; 
mC  =  elevation  of  line  of  sight  due  to  refraction: 

then 

AC  +  BC  _  2R  +  h  +  h'  =  tan  %(ABC  +  BAC) 
AC  -  BC  ~        h-h'         =  tan  ±(ABC  -  BAG)' 

ABC  +  BAC  =  180°  -  C, 

(c\        c 
90°  --j  =  cot-. 

ABC  =  180°  -  Z  -     mC 
BAC  =  Z  +    mC  -  C 


ABC  -  BAC  =  180°  -2Z  -2mC  +  C 
±(ABC  -  BAC)  =  90°  -  (z  +  mC  - 

tan  ±(ABC  -  BAC)  =  cot(z  +  mC  -  -  Y 


eot(z  +  mC  -r^J       tan  ^  cot  (Z  +  mC  -=-, 
\  /  \ 

C       /  C\ 

h  -h'  =  (2R  +  h  +  h')  tan  -  cotfZ  +  mC  -  -J. 

Expanding  tan  -  in  series,  we  have 
z 

C      C  .   C3  . 


GEODETIC  LEVELING  135 


. 

But  G  (in  arc;  =  -~  , 

whence 

C       K         K* 


Hence  by  substitution  and  reduction  and  the  omission  of  an  inap- 
preciable factor,  we  have 


Also 


C  (in  seconds)  =  ^— — 


R  sin  1 
whence 


=  K  tan  [«  +  (i  -  «)^-p]  (l  +  ~^  +  — 
or  approximately  (error  seldom  over  1  in  3000) 

h  —  hf  =  K  cot   Z  +  (m  —  J)p— "  ^7-,        (approximate), 

xL  sin  j- 

=  -K  tan   a  +  (J  —  m)  ^— ! — -7-,        (approximate). 
/t  sin  1 

The  value  of  (/i  —  h'}  is  always  found  first  by  the  approximate 
formula,  after  which  a  closer  value  may  be  obtained  from  the 
complete  formula  if  so  desired.  In  these  formulas  h,  h',  K,  and 
R  must  all  be  in  the  same  unit.  The  coefficient  of  refraction 
m  will  average  about  0.070  inland,  and  about  0.078  on  the  coast. 
The  radius  R  is  to  be  taken  for  the  middle  latitude  of  A  and  B 
and  the  approximate  azimuth  of  the  line  joining  them;  this 
value  may  be  taken  from  Tables  V  and  VI,  or  computed  as 
explained  in  Art.  fi9.  If  errors  which  may  reach  or  possibly 
exceed  about  1  in  500  are  permissible  we  may  use  a  mean  value 
of  R  and  write 

[metric,  6.8039665 ] 
logjR=    (feet,       7.3199507  jmeanvalue' 


136  GEODETIC  SURVEYING 

84.  By  Reciprocal  Observations.  When  the  distance  between 
two  stations  is  known  their  difference  of  elevation  can  be  com- 
puted without  assuming  any  particular  value  for  the  coefficient 
of  refraction  if  the  vertical  angle  of  each  station  as  seen  from 
the  other  is  measured.  This  result  is  brought  about  by  assuming 
that  the  refraction  is  the  same  at  each  station,  which  is  probably 
very  nearly  true  if  the  observations  are  made  at  the  same  time 
on  a  calm  day,  although  this  is  not  always  done.  The  advantage 
of  this  method  over  the  single  observation  method  (Art.  83) 
lies  in  the  increased  accuracy  of  the  results.  Fig.  42  (as  in  Fig.  41, 
Art.  83)  represents  a  plane  through  the  two  stations  A  and  B, 
taken  vertical  at  their  middle  latitude  and  assumed  to  be 


vertical  at  both  stations;  SS  is  the  elliptic  arc  cut  from  the 
spheroid,  but  it  is  here  assumed  to  be  the  arc  of  a  circle;  the 
radius  of  the  arc  SS  is  taken  as  the  mean-sea-level  radius  of  the 
section  at  the  middle  latitude,  the  center  C  being  in  general 
not  at  the  center  of  the  earth;  AC  and  BC  are  drawn  to  the 
center  C  and  assumed  to  be  vertical;  Z  and  Z'  are  the  apparent 
zenith  distances  of  the  stations  as  seen  from  each  other,  each 
angle  being  assumed  equally  in  error  by  the  small  angle  mC 
due  to  refraction. 

Let     h  =  AM  =  elevation  of  station  A  above  mean  sea  level; 
h'  =  BN  =  elevation  of  station  B  above  mean  sea  level ; 


GEODETIC  LEVELING  137 

K  =  MN  =  mean-sea.-level    distance    between    stations 

A  and  B-, 
R  =  MC  =  mean-sea-level  radius  of  section  at  middle 

latitude  between  A  and  B; 
C  =  central  angle  ACB; 

Z  =  apparent  zenith  distance  of  A  as  seen  from  B\ 
Zf  =  apparent  zenith  distance  of  B  as  seen  from  A ; 
«  =  90°  —  Z   =  apparent  elevation  of  A  as  seen  from  B; 
a'  =  90°  —  Z'  =  apparent  elevation  of  B  as  seen  from  A; 
mC  =  elevation  of  lines  of  sight  due  to  refraction; 
then, 

AC  +  BC  _  2R  +  h  +  h'  _  tan  %(ABC  +  BAG) 
AC  -BC  h  -h'  tan  %(ABC  -  BAG)' 

ABC  +  BAG  =  180°  -  C, 

tan  ±(ABC  +  BAG)  =  tan/W  -  ^  =  cot  £, 

\  A)  Z 

ABC  =  180°  -  Z    -mC 

BAG  =  180°  -  Z'  -  mC 
ABC  -  BAG  =  Z'  -  Z 

tan  i  (ABC  -  BAG)  =  tan  \(Z'  -  Z), 

,C 
2R  +  h  +  hf  !0t  2  1 


h  —  h'  tan  \(Z'  —  Z)  C 

tan  —  tan  \(Zr  —  Z) 

h  -  h'  =  (2R  +  h  +  h')  tan  -  tan  \(Z'  -  Z). 

C 
Expanding  tan  —  in  series,  we  have 

C  _  C      C3 
But 

n    f  \  K- 

C  (m  arc)  =  •_•  , 

whence 

'        K         K* 


138  GEODETIC  SURVEYING 

Hence  by  substitution  and  reduction  and  the  omission  of  an 
inappreciable  factor,  we  have, 

\  2# 

=  K  tan  i  (a  -  a')  1 1  +      2R      +  12R2) ' 

or  approximately  (error  seldom  over  1  in  3000) 

h  —  h'  =  K  tan  |  (Zf  —  Z)      (approximate), 
=  K  tan  %  (a  —  af)     (approximate) . 

The  value  of  (h  —  h')  is  always  found  first  by  the  approximate 
formula,  after  which  a  closer  value  may  be  obtained  from  the 
complete  formula  if  so  desired.  In  these  formulas  h,  h',  K, 
and  R  must  all  be  in  the  same  unit.  Except  for  very  important 
work  the  mean  value  of  R  as  given  in  Art.  83  is  sufficiently  precise. 
For  very  exact  results  the  radius  R  is  to  be  taken  for  the  middle 
latitude  of  A  and  B  and  the  approximate  azimuth  of  the  line 
joining  them;  this  value  may  be  taken  from  Tables  V  and  VI, 
or  computed  as  explained  in  Art.  69. 

85.  Coefficient  of  Refraction.  If  the  distance  between  two 
stations  is  known,  the  coefficient  of  refraction  m,  may  be  obtained 
as  follows: 

1st.  If  the  angular  elevation  of  either  station  as  seen  from 
the  other  is  measured,  and  the  difference  of  elevation  is  obtained 
by  spirit  leveling,  we  have  from  Art.  83, 

h  +  hf       J&_\ 
2R       r  VHP) 

~2R 12R2 

in  either  of  which  expressions  it  is  only  neccessary  to  substitute 
the  known  values  and  solve  for  m.  The  exact  value  of  R  is  to 
be  used,  as  explained  in  Art.  83. 

2nd.  If  the  angular  elevation  of  each  station  as  seen  from 
the  other  is  measued,  we  have  from  Fig.  42,  page  136, 

Z  +  mC  -  C  =  180°  -  Z'  -  mC] 
whence  2mC  =  180°  -  Z  -  Z'  +  C, 

180°  -  Z  -  Zf  +  C        a  +  a'  +  C 
and  m  =  —         —^ =  ^-^ , 


GEODETIC  LEVELING  139 

in  which  all  the  angular  values  must  be  expressed  in  the  same 
unit  (degrees,  minutes^  or  seconds).     From  Art.  83  we  have 

.     ..:••;•  •-.,•.;•'        C  (in  seconds)  =          —  , 


in  which  K  and  R  must  be  in  the  same  unit. 

The  average  value  of  the  coefficient  of  refraction  from  many 
Coast  Survey  observations  (Appendix  No.  9,  Report  for  1882), 
is  as  follows: 

Across  parts  of  the  sea  near  the  coast  ........  0.078 

Between  primary  stations  ..................  0.071 

In  the  interior  of  the  country  ..............  0.065 

86.  Accuracy  of  Trigonometric  Leveling.     The  U.  S.  Coast 
and  Geodetic  Survey  has  done  a  large  amount  of  leveling  of  this 
class   in    connection   with   its    triangulation   work,    with    sights 
sometimes  exceeding  a  hundred  miles  in  length  in  mountainous 
regions.     The  best  results  are  obtained  by  reciprocal  observations, 
taken  on  a  number  of  different  days  so  as  to  average  up  the 
atmospheric  conditions.     When  the  work  is  conducted  in  this 
manner  on  lines  not  over  about  20  miles  in  length  the  probable 
error  may  be  kept  down  to  about  one  inch  per  mile.     When  the 
lines  exceed  about  20  miles  in  length  it  is  necessary  to  take  a 
great  many  observations  under  especially  favorable  conditions  to 
secure  good   results.     In  order  to  prevent  an  accumulation  of 
errors    in  the  elevations  determined  by  trigonometric  leveling, 
connection  is  made  at  various  points   with  precise-level  bench 
marks,  and  the  trigonometric  leveling  is  adjusted  to  fit  the  precise 
leveling  between  these  points. 

C.     PRECISE  SPIRIT  LEVELING. 

87.  Instrumental  Features.     The  instruments  used  for  precise 
leveling  are  the  same  in  principle  as  the  various  types  of  engineers' 
levels,  the  essential  feature  being  a  telescopic  line  of  sight  and 
a  spirit  level  (detachable  or  fixed)  to  determine  its  horizontality. 
Engineers'  levels  are  designed  to  be  as  rapid  and   convenient 
in  use  as  possible,  consistent  with  the  requirements  of  engineering 
work.     Precise  levels  are  designed  to  attain  the  highest  possible 


140  GEODETIC  SURVEYING 

degree  of  precision  in  the  work  which  is  done  with  them.  Such 
instruments  are  made  in  various  forms,  two  of  which  are  shown 
in  Figs.  43  and  44  and  described  in  Arts.  89  and  90.  Certain 
features  are  more  or  less  common  to  all  types  of  precise  level. 
A  rigid  construction  and  the  highest  grade  of  material  and 
workmanship  are  demanded.  Especial  care  is  taken  to  make 
the  line  of  collimation  true  for  all  distances.  The  telescope  is 
made  inverting  (the  increased  illumination  permitting  a  higher 
magnifying  power),  and  has  three  horizontal  hairs  (as  equally 
spaced  as  possible)  whose  mean  position  determines  the  line  of 
sight.  The  convenience  of  having  the  line  of  sight  at  right 
angles  to  the  vertical  axis  of  the  instrument  is  abandoned  in 
order  to  place  a  delicate  control  of  the  position  of  the  bubble 
in  the  hands  of  the  observer;  this  is  accomplished  by  pivoting 
the  telescope  near  the  object-glass  end,  and  providing  a  fine 
screw  motion  near  the  eyepiece  end,  so  that  the  inclination  of 
the  telescope  can  be  changed  as  desired.  Such  a  screw  is  commonly 
called  a  micrometer  screw  because  it  was  originally  provided 
with  a  graduated  head  for  measuring  the  value  of  small  changes 
of  inclination.  The  level  vial  is  placed  above  the  telescope, 
and  a  mirror  or  other  means  provided  to  enable  the  observer 
to  see  the  bubble  at  the  moment  of  taking  an  observation.  A 
sensitive  bubble  is  used,  one  division  corresponding  to  about 
1  to  3  seconds  of  arc  (against  about  20  seconds  in  the  ordinary 
wye  or  dumpy  level).  The  level  vial  is  chambered,  permitting 
the  observer  to  adjust  the  bubble  to  its  most  efficient  length, 
and  is  so  mounted  that  it  is  free  to  expand  and  contract.  The 
instrument  is  supported  on  three  pointed  leveling  screws  resting 
freely  in  V-shaped  metal  grooves  on  the  tripod  head.  Such 
an  instrument  is  leveled  by  setting  the  bubble  parallel  to  a  pair 
of  leveling  screws  and  bringing  it  to  the  center  by  turning  that 
pair  of  screws  equally  in  opposite  directions,  then  turning  the 
bubble  in  line  with  the  remaining  leveling  screw  and  bringing 
it  to  the  center  with  that  screw  alone;  then  turn  the  instrument 
180°  on  its  vertical  axis,  and  if  the  bubble  moves  from  the  center 
bring  it  half  way  back  by  the  micrometer  screw  of  the  telescope 
and  relevel  both  ways  as  before;  when  the  bubble  will  stay  within 
a  few  divisions  of  the  center  all  the  way  around  the  leveling 
is  satisfactory,  as  the  precise  leveling  of  the  line  of  sight  is  accom- 
plished with  the  micrometer  screw  while  taking  the  observation. 


GEODETIC  LEVELING 


141 


142 


GEODETIC  SURVEYING 


GEODETIC  LEVELING  143 

The  tripods  used  with  these  instruments  must  be  strong  and  rigid. 
Rods  of  special  pattern  and  metallic  turning  points,  as  described 
in  Art.  91,  are  used  in  this  class  of  work. 

88.  General  Field  Methods.  In  order  to  secure  a  high  degree 
of  precision  in  leveling  the  greatest  care  is  required  in  the  field 
work  and  methods.  Five  sources  of  error  have  to  be  guarded 
against,  namely,  errors  of  observation,  instrumental  errors, 
curvature  and  refraction  errors,  atmospheric  errors,  and  errors 
from  unstable  supports. 

Errors  of  observation  are  kept  as  small  as  possible  by  care 
on  the  part  of  the  observer;  by  keeping  the  rods  plumb;  by 
using  a  proper  length  of  sight,  100  meters  or  about  300  feet 
being  suitable  for  average  conditions;  by  comparing  at  every 
sight  the  two  intervals  furnished  by  the  readings  of  the  three 
wires,  any  material  disagreement  (more  than  2  millimeters) 
denoting  an  erroneous  reading;  by  the  fact  that  each  pointing 
is  taken  as  the  mean  of  the  three  wire  readings;  and  by  the 
further  fact  that  every  line  is  run  in  duplicate  in  the  reverse 
direction  and  a  limit  set  on  the  allowable  discrepancies. 

Instrumental  errors  are  kept  as  small  as  possible  by  keeping 
the  instrument  in  good  adjustment;  by  determining  the  instru- 
mental constants  with  care  and  applying  the  corresponding 
corrections  when  necessary;  by  using  a  program  of  observations 
adapted  to  the  type  of  instrument  used,  so  as  to  eliminate  the 
instrumental  errors  as  far  as  possible;  by  making  the  length 
of  each  foresight  nearly  equal,  if  possible,  to  that  of  the  corre- 
sponding backsight;  by  balancing  any  extra  long  or  short  fore- 
sight by  a  similar  long  or  short  backsight  elsewhere,  and  vice 
versa;  and  by  keeping  the  sum  of  the  lengths  of  the  foresights 
as  nearly  equal  as  possible  to  the  sum  of  the  lengths  of  the  back- 
sights, with  suitable  corrections  for  the  net  difference.  If  the 
foresights  and  backsights  were  all  exactly  equal  no  correction 
would  be  required  for  instrumental  errors.  The  effect  of  the 
various  instrumental  errors  is  to  give  the  line  of  sight  an  inclina- 
tion with  the  horizontal.  The  value  of  the  inclination  becomes 
known  through  the  instrumental  constants,  as  explained  later. 
The  required  correction  in  elevation  is  found  by  multiplying 
the  net  difference  in  length  of  sights  by  the  sine  of  this  incli- 
nation. 

Curvature  and  refraction  errors  exist  in  every  line  of  sight, 


144  GEODETIC  SURVEYING 

as  explained  in  Art.  14,  but  are  obviously  eliminated  if  the 
foresights  and  backsights  are  kept  equal.  If  these  sights  are 
kept  nearly  balanced,  as  explained  in  the  previous  paragraph, 
and  a  suitable  correction  made  for  the  net  difference,  the  effects 
of  curvature  and  ordinary  refraction  are  practically  reduced  to 
zero.  The  correction  which  is  made  is  the  value  of  the  curva- 
ture and  refraction  for  the  net  difference  in  the  lengths  of  the 
foresights  and  backsights.  The  net  difference  should  be  kept 
so  small  that  no  such  correction  may  be  necessary,  but  if  required 
it  can  be  taken  from  Table  VII  or  computed  as  explained  in 
Art.  14. 

Atmospheric  errors  are  those  due  to  an  actual  unsteadiness 
of  the  rod  or  instrument,  caused  by  the  wind;  an  apparent 
unsteadiness  of  the  rod,  caused  by  heated  air  currents,  commonly 
called  heat  radiation;  an  irregular  vertical  displacement  of  the 
line  of  sight,  caused  by  variable  refraction;  and  the  disturbance 
of  the  relation  between  the  line  of  sight  and  the  axis  of  the  bubble, 
caused  by  unequal  expansion  and  contraction  of  the  different 
parts  of  the  instrument.  Moderate  winds  do  not  prevent  good 
work,  especially  if  wind  shields  are  used  around  the  instrument; 
but  when  the  wind  reaches  about  eight  miles  an  hour  it  becomes 
impracticable  to  do  first  class  work.  When  the  rod  becomes  un- 
steady through  heat  radiation  it  becomes  necessary  to  decrease 
the  length  of  the  sights  in  order  to  read  the  rod  satisfactorily,  but 
the  increased  number  of  sights  increases  the  probable  error  of 
the  result;  if  it  becomes  necessary  to  decrease  the  length  of  sight 
below  50  meters,  or  about  150  feet,  it  is  not  advisable  to  continue 
the  work.  Refraction  is  nearly  stationary  and  has  its  least 
value  between  about  9.00  A.M.  and  3.30  P.M.,  but  during  this 
period  heat  radiation  is  apt  to  be  very  troublesome;  outside 
of  these  hours  the  refraction  may  be  very  variable.  The  result 
is  that  in  perfectly  clear  weather  the  best  class  of  work  is  only 
possible  during  a  few  hours  of  the  day.  In  order  to  guard  against 
unequal  expansion  and  contraction  the  instrument  is  protected 
with  a  large  sunshade  (umbrella),  and  never  exposed  to  the  direct 
rays  of  the  sun  either  while  in  use  or  while  being  carried  to  a  new 
set-up. 

By  the  errors  from  unstable  supports  are  meant  the  errors 
caused  by  the  instrument  or  turning  points  changing  their  eleva- 
tions slightly  between  readings.  It  is  shown  by  experience  that 


GEODETIC  LEVELING  145 

either  rising  or  settling  may  take  place,  though  settling  is  the 
most  common.  If  the  instrument  settles  between  the  backward 
reading  and  the  forward  reading  the  final  elevation  will  be  too 
high;  the  same  result  will  occur  if  the  rod  settles  between  the 
forward  reading  and  the  backward  reading  on  it.  Errors  of  this 
class  are  kept  as  small  as  possible  by  planting  the  instrument 
firmly;  by  using  well  driven  metallic  turning  points;  by  taking 
both  readings  from  each  set-up  with  as  little  intermediate  delay 
as  possible,  using  two  rodmen  for  this  reason  as  well  as  the  saving 
of  time;  by  reading  the  back  rod  first  for  every  other  set-up, 
and  the  fore  rod  first  for  the  intermediate  set-ups;  and  by  duplicat- 
ing each  line  in  the  opposite  direction,  and  correcting  for  half 
of  the  discrepancy. 

Certain  field  methods  have  been  discarded,  after  years  of 
extensive  use,  because  the  results  have  not  proven  as  satisfactory 
as  by  other  methods.  Among  these  may  be  mentioned  methods 
involving  computations  based  on  readings  of  the  micrometer 
screw.  The  best  results  are  obtained  when  all  the  observations  are 
taken  with  the  bubble  in  the  center,  the  micrometer  screw  being 
used  simply  as  the  means  of  keeping  it  there.  Another  unsatis- 
factory method  is  the  running  of  so-called  simultaneous  lines, 
in  which  readings  are  taken  at  each  set-up  to  the  turning  points 
of  two  separate  lines,  as  a  substitute  for  running  duplicate  lines 
in  opposite  directions. 

89.  The  European  Level.  A  typical  leveling  instrument  of 
this  form,  made  in  France,  is  illustrated  in  Fig.  43  (page  141). 
The  European  type  of  instrument  is  essentially  a  wye  level, 
in  which  different  makers  have  followed  the  same  general 
design,  but  with  modified  details.  The  telescope  may  be  rotated 
in  the  wyes  or  lifted  from  the  wyes  and  reversed.  The  level 
is  separate  from  the  instrument,  being  an  ordinary  striding 
level  with  the  addition  of  a  movable  mirror  over  the  bubble; 
by  holding  the  eyes  in  a  vertical  line  the  image  of  the  bubble 
may  be  seen  with  one  eye  while  the  rod  is  seen  through  the  tele- 
scope with  the  other  eye,  the  bubble  being  kept  in  the  center 
with  the  micrometer  screw  while  the  observation  is  being  made. 
The  magnifying  power  is  about  forty-five  diameters.  Besides 
the  above  special  features  the  instrument  has  all  the  general  features 
of  a  good  instrument,  as  described  in  Art.  87.  With  this  type  of 
level  there  are  three  so-called  constants  and  two  adjustments. 


146  GEODETIC  SURVEYING 

89a.  Constants  of  European  Level.  The  three  constants  of 
this  instrument,  which  should  be  examined  at  least  once  a  year, 
are  as  follows: 

1.  The  angular  value  of  one  division  of  the  bubble,  meaning  the 
change  in  inclination  which  causes  the  bubble  to  shift  its  position 
by  one  division  on  the  bubble  scale.  Modern  level  vials  are 
ground  so  .nearly  uniform  in  curvature  that  it  is  customary  to 
measure  the  change  of  inclination  for  the  whole  run  of  the  bubble, 
dividing  by  the  number  of  divisions  through  which  the  bubble 
moves  to  obtain  the  average  value  of  one  division.  By  the  posi- 
tion of  the  bubble,  or  the  movement  of  the  bubble,  is  meant  the 
position  or  the  movement  of  its  central  point;  the  ends  of  the 
bubble  are  constantly  changing  their  position  on  account  of  the 
changing  length  of  the  bubble,  but  the  center  remains  stationary 
as  long  as  there  is  no  change  of  inclination.  Bubble  tubes  are 
sometimes  graduated  from  one  end,  but  more  frequently  both 
ways  from  the  center,  in  which  case  the  divisions  one  way  from 
the  center  are  called  positive  and  the  other  way  negative.  The 
reading  of  the  center  of  the  bubble  is  the  algebraic  mean  of  its 
two  end  readings.  The  movement  of  the  bubble  between  any 
two  positions  is  the  algebraic  difference  of  its  two  center  readings. 
The  practical  operation  of  finding  the  value  of  one  division  is  as 
follows:  Level  up  the  instrument  with  the  striding  level  in  place, 
and  have  a  leveling  rod  held  at  a  fixed  point  at  a  known  distance 
of  about  200  feet.  Turn  the  micrometer  screw  until  the  bubble 
comes  near  one  end  of  its  run,  note  each  wire  reading  on  the  rod 
as  closely  as  possible,  and  each  end  reading  of  the  bubble  to  the 
nearest  tenth  of  a  division.  Run  the  bubble  to  the  other  end 
of  the  tube  and  note  the  rod  and  bubble  readings  for  this  posi- 
tion. Take  a  number  of  readings  in  this  way  at  both  ends,  with 
the  bubble  in  slightly  different  positions  so  as  to  obtain  unbiassed 
values.  Compute  the  position  of  the  center  of  the  bubble  for  each 
reading,  then  the  average  of  the  center  readings  for  each  end  of 
the  run,  and  then  the  movement  corresponding  to  these  average 
centers,  which  will  be  the  average  movement  of  the  bubble. 
Subtract  the  mean  of  the  lower  readings  from  the  mean  of  the 
upper  readings  on  the  rod  for  the  average  movement  of  the  line 
of  sight,  which  divided  by  the  distance  times  the  sine  of  1"  will 
give  the  average  change  of  inclination  in  seconds  of  arc.  The 
angular  value  of  one  division  of  the  bubble  in  seconds  will  be  this 


GEODETIC  LEVELING 


147 


average  change  of  inclination  divided  by  the  average  movement 
of  the  bubble.  In  this  process  the  rod  readings  and  the  dis- 
tances must  be  expressed  in  the  same  unit.  In  the  following 
example  illustrating  the  above  principles  the  bubble  tube  is 
graduated  each  way  from  the  center  and  a  metric  rod  is  held  70 
meters  from  the  instrument.  Each  recorded  rod  reading  is  the 
average  of  the  three  wire  readings. 


EXAMPLE, — ANGULAR  VALUE  OF  ONE  DIVISION  OF  BUBBLE  TUBE 


Looking  Up. 

Looking  Down. 

Rod. 

Bubble. 

Rod. 

Bubble. 

Left. 

Right. 

Center. 

Left. 

Right. 

Canter. 

1.5245 

-36.9 

+  3.1 

-16.9 

1.5000 

-1.5 

+  37.8 

+  18.2 

1.5240 

-36.1 

+  2.8 

-16.7 

1.5005 

-1.7 

+  36.7 

+  17.5 

1.5245 

-36.0 

+  2.0 

-17.0 

1.5010 
1.5005 

-2.0 
-1.4 

+  35.4 
+  35.9 

+  16.7 

1.5250 

-35.6 

+  1.1 

-17.3 

+  17.2 

1.5245 

-35.8 

+  1.8 

-17.0 

1.5005 

-1.5 

+  35.7 

+  17.1 

7.6225 
1.5245 

Sums 

-84.9 

7.5025 

Sums 

+  86.7 

Means 

-16.98 

1.5005 

Means 

+  17.34 

sin  1"=  0.000004848 
70  X  sin  1"  =  0.00033936 
0  .  0240  -*-  0  .  00033936  =  70"  .  72 
Change  of  inclination  =70".  72 

1  .  5245 
0.0240 

Algebraic 
differences 

-16.98 
34.32 

70".  72  -=-34.32=  2".  06 

Angular  value  of  one  di  vision  =  2".  1 

2.  The  inequality  of  the  pivot  rings,  meaning  the  angle  between 
the  line  joining  the  tops  of  the  pivot  rings  (the  telescope  collars 
that  rest  in  the  wyes)  and  the  center  line  of  these  rings.  This 
angle  would  of  course  be  zero,  if  there  were  no  inequality  in  the 
size  of  the  rings;  but  a  small  angle  generally  exists,  due  usually 
to  unequal  wear.  It  follows  that  when  the  tops  of  the  rings  are 
in  a  level  plane,  as  indicated  by  the  striding  level,  the  line  of 
sight  or  center  line  of  the  rings  must  be  inclined  to  the  horizontal 
to  the  extent  of  this  angle.  In  order  to  determine  this  value 


148  GEODETIC  SURVEYING 

the  instrument  is  approximately  leveled,  and  clamped  on  its 
vertical  axis.  Bubble  readings  are  then  taken  with  the  telescope 
direct  and  also  when  reversed  end  for  end  in  the  wyes.  If  the 
striding  level  and  telescope  were  reversed  together  (as  one  piece) 
the  movement  of  the  bubble  would  measure  twice  the  angle 
between  the  axis  of  the  bubble  and  the  bottom  line  of  the  pivot 
rings.  If  the  striding  level  were  in  perfect  adjustment  (axis  of 
bubble  parallel  to  line  of  feet)  this  would  mean  the  same  thing 
as  twice  the  angle  between  the  top  line  and  bottom  line  of  the 
rings,  or  four  times  the  pivot  inequality  (angle  between  center 
line  and  tops  of  rings).  The  striding  level  is  seldom  in  perfect 
adjustment,  but  its  error  is  eliminated  by  taking  its  average 
reading  for  its  direct  and  reversed  positions  for  each  position 
of  the  telescope.  The  telescope  is  generally  reversed  a  number 
of  times  and  the  average  result  taken.  It  is  found  in  practice 
that  the  inclination  of  the  telescope  is  liable  to  be  changing 
during  the  progress  of  the  observations,  and  thus  lead  to  erroneous 
conclusions.  Readings  are  therefore  not  only  taken  for  alternate 
positions  of  the  telescope,  but  the  last  position  is  made  the  same 
as  the  first  position;  the  assumption  is  then  made  that  the  mean 
of  the  direct  sets  and  the  mean  of  the  reverse  sets  correspond 
to  the  same  instant  of  time.  When  the  pivot  inequality  is 
obtained  in  bubble  divisions  its  angular  value  is  found  by  multiply- 
ing this  result  by  the  angular  value  of  one  division  of  the  bubble. 
In  the  following  example  illustrating  the  above  principles  the 
level  tube  is  graduated  both  ways  from  the  center,  and  is  called 
direct  with  the  marked  end  towards  the  eyepiece. 

It  will  be  noted  in  this  example  that  the  average  effect 
of  reversing  the  telescope  (from  eye-end  left  to  eye-end  right), 
is  to  cause  the  bubble  to  move  to  the  right  or  towards  the  eye- 
end,  showing  the  eye-end  ring  to  be  larger  than  the  other  ring 
which  it  replaces;  when  the  tops  of  the  rings  are  in  a  level  plane, 
therefore,  as  indicated  by  the  striding  level,  it  follows  that  the 
line  of  sight  (center  line  of  the  rings)  must  look  up.  If  the  tele- 
scope looks  up  it  will  cause  the  final  elevation  to  be  too  low  for 
an  excess  in  the  foresights  and  too  high  for  an  excess  in  the  back- 
sights, and  vice  versa  when  the  telescope  looks  down.  The 
amount  of  the  correction  required  will  be  equal  to  the  excess 
distance  multiplied  by  the  angular  inequality  of  the  pivots  and 
by  the  sine  of  1". 


GEODETIC  LEVELING 


149 


EXAMPLE. — INEQUALITY  OF  PIVOT  RINGS 

m 


Telescope. 

Level. 

Bubble  Readings. 

Left. 

Right. 

Left. 

Right. 

Eye-end  left 

Direct 
Reversed 

-.  26.6 

-   28.0 

+  23.7 
+  22.4 

right  .... 

Direct 
Reversed 

-24.0 
-25.4 

+  26.5 
+  25.0 

left    

Direct 
Reversed 

-  26.9 

-   28.4 

+  23.8 
+  22.4 

right  .... 

Direct 
Reversed 

-24.2 
-25.8 

+  26.8 
+  25.2 

left  

Direct 
Reversed 

-  27.2 

-  28.6 

+  24.1 
+  22.8 

Sums 

-165.7 

+  139.2 

-99.4 

+  103.5 

Means 

-  27.62 

+  23.20 

-24.85 

+  25.88 

Center  of  bubble 

-2.21 

+  0.52 

Eye  -end  ring  large 

Bubble  moves  to  right    =  2  .  73  div. 

Telescope  looks  up 

Inequality  of  pivots    =—0.68    " 

0.68X2".  1=1".  428 

Ang,  inequality  of  pivots=  —  1".4 

3.  The  angular  value  of  the  wire  interval,  meaning  the  ratio 
between  the  solar  focus  or  principal  focal  length  of  the  objective 
and  the  distance  between  the  outer  cross-hairs.  The  telescope 
may  be  regarded  as  set  for  a  solar  focus  when  it  is  focussed  on  any 
distant  object. 

Let  D  =  unknown   distance  between  level   rod   and  vertical 

axis  of  level ; 

S  =  corresponding  rod  intercept  between  outer  cross-hairs; 
d  =  a  known  distance  from  axis  of  level; 
s  =  corresponding  intercept; 

/  =  distance  from  cross-hairs  to  objective  for  solar  focus; 
c  =  distance  from  vertical  axis  to  objective  for  solar  focus; 
i  =  distance  between  outer  cross-hairs; 

A  =  -=  angular  value  of  wire  interval; 
^ 


150  GEODETIC  SURVEYING 

then,  from  the  theory  of  stadia  measurements, 


A  -  f  - 

=  ~  " 


D  =  A-S  +  (/+c), 

in  which  formulas  D,  S,  d,  s,  f,  and  c  must  all  be  taken  in  the  same 
unit.  The  field  work  of  finding  A  consists  in  focussing  on  a 
distant  point  and  measuring  on  the  telescope  the  values  of  / 
and  c;  then  measure  a  distance  of  about  100  meters  or  about  300 
feet  from  the  vertical  axis  of  the  instrument,  and  take  the  rod 
readings  at  this  point  (with  the  instrument  leveled)  for  the  upper 
and  lower  hairs;  the  intercept  s  of  the  formula  is  the  difference 
of  these  readings;  then  substitute  the  values  d,  s,  /,  and  c  in  the 
formula  for  A.  The  value  of  A  may  run  from  about  100  to  about 
300,  the  instrument  maker  usually  setting  the  hairs  as  near  as 
possible  for  an  even  hundred.  With  the  value  of  A  known  and 
the  recorded  rod  readings  a  simple  substitution  in  the  formula 
for  D  at  once  gives  the  distance  between  the  instrument  and  cor- 
responding turning  point.  Since  the  corrections  for  instrumental 
errors  are  only  applied  to  the  excess  distance  between  foresights 
and  backsights,  a  running  total  is  kept  of  the  corresponding 
wire  intervals,  and  the  formula  for  D  applied  to  this  excess  interval 
only,  omitting  the  small  constant  (/+c). 

89b.  Adjustments  of  European  Level.  The  two  adjustments 
of  this  instrument,  which  should  be  examined  daily,  are  as  follows  : 

1.  The  collimation  adjustment,  meaning  the  adjustment  of 
the  position  of  the  ring  that  carries  the  cross-hairs  so  that  the 
actual  line  of  sight  (as  indicated  by  the  mean  position  of  the  hairs) 
shall  coincide  with  the  true  line  of  sight  or  center  line  of  the  rings. 
This  adjustment  is  made  by  leveling  up  the  instrument  and  sight- 
ing at  a  rod  (about  100  meters  distant)  with  the  telescope  both 
direct  and  inverted.  If  the  mean  of  the  three  wire  readings  is 
not  the  same  in  each  case  the  reticule  is  moved  in  the  apparent 
direction  needed  to  correct  the  error  and  an  amount  equal  to 
half  the  discrepancy.  It  is  essential  that  the  instrument  be 
perfectly  leveled  for  each  reading.  When  the  discrepancy  is 
brought  down  to  about  two  millimeters  it  may  be  considered 
satisfactory,  as  it  is  easy  to  apply  a  correction  for  the  residual 


GEODETIC  LEVELING  151 

error,  or  the  error  may_be  elimiiiated  by  the  method  of  observing. 
The  collimation  error  is  the  angular  amount  by  which  the 
actual  line  of  sight  (determined  by  mean  position  of  cross-hairs) 
deviates  from  the  center  line  of  the  rings.  The  collimation  error 
only  affects  the  excess  distance,  like  all  the  other  instrumental 
errors.' 

Let    C  =  collimation  correction  for  excess  distance  D; 

D  =  excess  distance  between  backsights  and  foresights; 
c  =  collimation  error; 
d  =  a  known  distance; 

RI  =  mean  rod  reading  for  d  with  telescope  normal; 
RZ  =  mean  rod  reading  for  d  with  telescope  inverted; 

then  evidently, 

D  E> 

and    C  =  cD, 


2.i 

in  which  all  values  must  be  taken  in  the  same  unit. 

2.  The  bubble  adjustment,  meaning  the  adjustment  by  which 
the  axis  of  the  bubble  is  made  parallel  to  the  line  joining  the  feet 
of  the  striding  level.  This  adjustment  is  made  by  leveling  up 
the  instrument,  clamping  the  vertical  axis,  bringing  the  bubble 
exactly  central  with  the  micrometer  screw,  and  then  reversing 
the  striding  level  without  disturbing  the  telescope.  If  the  bubble 
is  not  central  after  reversal  it  is  to  be  adjusted  for  one-half  of 
its  movement.  Relevel  with  the  micrometer  screw,  reverse 
again,  and  so  on  until  the  adjustment  is  satisfactory  (within 
about  one  division  of  the  scale).  The  bubble  error  or  inclination 
of  the  bubble  is  the  angle  between  the  axis  of  the  bubble  and  the 
line  joining  the  feet  of  the  striding  level;  this  angle  would  be 
zero  if  the  bubble  were  in  perfect  adjustment.  To  determine 
the  bubble  error  level  up  the  instrument  approximately,  clamp  the 
vertical  axis,  bring  the  bubble  near  the  center  with  the  micrometer 
screw,  and  then  read  the  bubble  a  number  of  times  in  direct  and 
reversed  positions,  making  the  last  position  the  same  as  the  first 
position.  The  bubble  error  in  bubble  divisions  is  half  the  average 
movement  of  the  bubble;  the  inclination  of  the  bubble  is  the  error 
in  bubble  divisions  multiplied  by  the  angular  value  of  one 
division.  In  the  following  example  illustrating  the  above  principles 
the  level  tube  is  graduated  both  ways  from  the  center,  and  is 
called  direct  with  the  marked  end  towards  the  eyepiece. 


152 


GEODETIC  SURVEYING 
EXAMPLE. — INCLINATION  OF  BUBBLE 


Eyepiece  to  the  left 

Striding  Level. 

Bubble. 

Left. 

Right. 

Left. 

Right. 

Direct 

-26.6 

+  23.7 

Reversed      .            .... 

-28.0 

+  22.4 

Direct       

-26.9 

+  23.8 

Reversed  

-28.4 

+  22.4 

Direct  

—  27.2 

+  24.1 

Sums 

-80.7 

+  71.6 

-56.4 

+  44.8 

Means 

-26.90 

+  23.87 

-28.20 

+  22.40 

Center  of  bubble 

-1.52 

-2.90 

Level  direct—  Telescope  looks  down. 

Bubble  error  =  +0.69  division. 

0.69X2".  1=1".  449 

Inclination  of  bubble  =  +  1".4 

It  will  be  noted  in  the  above  example  that  the  average  effect 
of  reversing  the  striding  level  (putting  marked  end  towards 
object  glass)  is  to  cause  the  bubble  to  move  away  from  the 
marked  end,  showing  that  the  marked  end  has  the  shorter  leg; 
when  the  bubble  is  in  the  center,  therefore,  if  the  marked  end  of 
the  striding  level  is  nearest  the  eyepiece  the  telescope  looks  down. 
If  a  line  of  levels  were  run  with  the  striding  level  in  a  fixed  posi- 
tion a  correction  would  be  required  for  the  excess  distance,  the 
value  of  which  would  equal  the  inclination  of  the  bubble  multiplied 
by  the  excess  distance  and  the  sine  of  1".  The  sign  of  the  cor- 
rection for  excess  of  foresights  would  be  positive  for  telescope 
looking  up  and  negative  looking  down,  and  vice  versa  for  excess 
of  backsights. 

89c.  Use  of  European  Level.  The  best  results  are  obtained 
when  all  the  rod  readings  are  taken  with  the  bubble  precisely 
centered,  and  the  observations  so  arranged  as  to  eliminate  as 
far  as  possible  the  effects  of  the  instrumental  errors.  All  the 
precautions  of  Art.  88  are  to  be  carefully  observed.  Among 
these  may  be  again  mentioned  the  necessity  of  keeping  the 
instrument  sheltered  by  the  umbrella  from  the  sun  and  wind  at 
all  times;  making  each  foresight  approximately  equal  to  the 


GEODETIC  LEVELING  153 

previous  backsight  (pacing  is  satisfactory);  keeping  the  sum  of 
the  foresights  nearly -equal  to  the  sum  of  the  backsights,  as 
indicated  by  the  corresponding  sums  of  the  wire  intervals;  plant- 
ing the  instrument  firmly  and  making  the  turning  points  solid; 
keeping  the  rod  plumb;  watching  the  wire  intervals  at  every 
sight,  and  taking  a  new  reading  of  each  of  the  three  wires  when- 
ever the  half  intervals  disagree  by  more  than  two  millimeters; 
and  running  a  duplicate  line  in  the  opposite  direction  as  a  check, 
and  in  order  to  eliminate  errors  from  unstable  supports  (by  using 
the  mean  difference  of  elevation  as  the  true  value). 

Program  of  observations  for  each  set-up.  Level  up  the  instru- 
ment; sight  at  the  back  rod;  take  each  of  the  three  wire  readings 
with  the  bubble  kept  centered  with  the  micrometer  screw;  sight 
on  the  forward  rod  and  read  with  bubble  central  as  before;  remove 
striding  level,  invert  telescope  in  wyes,  replace  striding  level 
reversed  end  for  end;  read  forward  rod  with  bubble  central; 
sight  on  back  rod  and  read  with  bubble  central.  This  method 
of  observing  eliminates  both  the  bubble  error  and  the  collimation 
error,  even  with  the  foresights  and  backsights  unbalanced.  The 
correction  for  inequality  of  pivots,  however,  must  be  applied  to 
any  excess  distance,  as  also  the  correction  for  curvature  and 
refraction  if  the  excess  distance  makes  the  amount  appreciable. 
An  example  of  notes  and  reductions  is  given  on  the  next  page. 
In  this  case  the  backsights  are  in  excess,  but  not  enough  to  require 
appreciable  corrections. 

90.  The  Coast  Survey  Level.  Previous  to  1900  the  precise 
leveling  of  the  U.S.  Coast  and  Geodetic  Survey  was  done  with 
the  European  type  of  instrument.  Commencing  with  the  summer 
of  1900  this  work  has  been  done  with  a  type  of  instrument  designed 
by  the  Department  and  known  as  the  Coast  Survey  level.  A 
view  of  this  level  is  shown  in  Fig.  44,  page  142.  The  instrument 
is  essentially  a  dumpy  level,  as  the  telescope  does  not  rest  in  wyes, 
can  not  be  removed  from  its  supports,  and  can  neither  be  inverted 
nor  reversed.  The  base  of  the  instrument  is  of  the  usual  three 
leveling  screw  type,  except  that  the  center  socket  is  unusually 
long  and  extends  downwards  through  the  tripod  head.  An 
outer  protecting  tube  through  which  the  telescope  passes  is 
rigidly  attached  to  the  vertical  axis;  the  telescope  is  pivoted  at 
one  end  of  this  outer  tube,  and  has  its  inclination  controlled  by  a 
micrometer  screw  at  the  other  end.  The  collimation  adjustment 


154 


GEODETIC  SURVEYING 


FORM  OF  NOTES— EUROPEAN  LEVEL 

(Left-hand  page.)  (Right-hand  page/ 


Forward  Line. 
Backsights. 

B.  M.  4  to  B.  M.  5. 
Date,  June  18,  1911. 

Point. 

Rod. 
and 
Temp. 

Thread  Readings. 

Mean. 

Intervals. 

Remarks. 

1 

2 

3 

Each. 

Sums. 

B.M.4 

r.  p.  i 

2 

76 

2.518 
2.522 

2.616 
2.618 

2.714 
2.716 

2.6170 

2.3967 
+  5.0137 

0.1950 
0.1625 

0.1950 
0.3575 

Elevation  of 
B.M.  4=117.  617 

/DescriptionX 
\ttfB.  M.4.J 

Means 

5 

78 

2.5200 

2.313 
2.318 

2.6170 

2.395 

2.398 

2.7150 

2.476 

2.480 

Means 

2.3155 

2.3965 

2.4780 

(Left-hand  page,)                                (Right-hand  page,) 

Forward  Line. 
Foresights. 

B.  M.  4  to  B.  M.  5. 
Date,  June  18,  1911. 

Point. 

Rod. 
and 
Temp. 

Thread  Readings. 

Mean. 

Intervals. 

Remarks. 

1 

2 

3 

Each. 

Sums. 

T.  P.  1 
B.M.5 

5 

77 

1.167 
1.173 

1.260 
1.266 

1.354 
1.361 

1.2635 

0.8008 

-2.0643 
+  5.0137 

0.1875 
0.1585 

0.1875 

0.3460 

0.3460 
0.3575 

/DescriptionX 
\  of  B.  M.  5./ 

Elevation  of 
B.M.  4=117.617 
+  2.949 

Means 

2 

78 

1.1700 

0.720 
0.723 

1.2630 

0.800 
0.802 

1.3575 

0.880 
0.880 

Means 

0.7215 

0.8010 

0.8800 

+  2.9494 

0.0115 

B.M.  5=120.566 

is  permanently  fixed  by  the  maker.  The  level  tube  is  attached 
to  the  telescope,  but  has  provision  for  adjustment.  A  strong 
point  of  the  instrument  is  the  closeness  of  the  bubble  to  the  line 
of  sight,  the  level  tube  being  let  part  way  into  a  slot  cut  in  the 
top  of  the  telescope  tube,  the  top  of  the  level  tube  coming  about 
flush  with  a  slot  in  the  top  of  the  outer  tube.  The  level  vial  is 
chambered  for  adjusting  the  length  of  the  bubble.  Attached  to 
the  left  side  of  the  instrument  is  a  light  auxiliary  tube  through 


GEODETIC  LEVELING  155 

which  the  left  eye  may  see  an  -.image  of  the  bubble  while  the  right 
eye  is  observing  the  Tod,  the  head  being  held  in  its  natural  posi- 
tion, and  the  tube  being  adjustable  sideways  to  suit  the  eyes 
of  different  observers.  Besides  the  lens  in  its  eyepiece  the  tube 
contains  two  prisms,  adjustable  for  length  of  bubble,  and  placed 
opposite  a  slot  running  abreast  of  the  level  vial.  The  bubble  is 
brought  within  the  view  of  the  left  eye  through  the  eye  lens,  the 
two  prisms,  and  a  mirror  attached  to  the  telescope.  The  telescope 
tube  and  outer  casing  are  made  of  a  nickel-iron  alloy  that  has 
a  coefficient  of  expansion  which  is  only  one-fourth  that  of  brass, 
while  the  micrometer  screw  and  other  important  screws  are  made 
of  nickel-steel  having  a  coefficient  of  expansion  as  low  as  0.000001 
per  degree  centigrade.  A  detailed  description  of  this  instrument 
(from  which  the  above  notes  have  been  gathered)  is  given  in 
Appendix  No.  3,  Report  for  1903,  U.  S.  Coast  and  Geodetic 
Survey.  Work  with  this  level  has  been  extremely  satisfactory, 
better  results  being  secured  with  greater  rapidity  and  a  much 
reduced  cost.  The  Coast  Survey  level  has  two  constants  and 
one  adjustment. 

90a.  Constants  of  Coast  Survey  Level.  The  two  constants  of 
this  instrument,  which  should  be  examined  at  least  once  a  year, 
are  as  follows : 

1.  The   angular   value  of  one   division  of  the  bubble.      This  is 
found  by  the  optical  method,  as  described  in  Art.  89a. 

2.  The    angular    value    of    the   wire    interval.     This    is    also 
found  as  described  in  Art.  89a. 

90b.  Adjustments  of  Coast  Survey  Level.  The  only  adjust- 
ment of  this  instrument,  which  should  be  examined  daily,  is  as 
follows : 

To  make  the  axis  of  the  bubble  parallel  to  the  line  of  sight. 
This  adjustment  is  made  by  the  ordinary  peg  method  (as  adapted 
to  this  type  of  instrument),  the  bubble  tube  being  raised  or  lowered 
at  the  adjusting  end  as  may  be  required.  The  cross-hairs  must 
never  be  disturbed  as  these  have  been  permanently  adjusted  for 
collimation  by  the  instrument  maker.  In  testing  the  adjustment 
the  rod  reading  is  taken  as  the  mean  of  the  three  wire  readings, 
and  the  rod  interval  as  the  difference  between  the  outside  wire 
readings,  the  bubble  being  kept  exactly  centered  while  reading 
each  of  the  three  wires.  Two  pegs  or  turning-point  pins  are 
firmly  driven  about  100  meters  apart,  each  rod  being  kept 


156  GEODETIC  SURVEYING 

on  its  own  point  if  two  rods  are  used,  or  one  rod  being  shifted  as 
required.  The  instrument  is  set  up  approximately  in  line  with 
the  two  points,  first  about  ten  meters  beyond  one  point,  and  then 
about  the  same  distance  beyond  the  other  point.  The  rod  read- 
ing is  taken  for  each  point  in  each  position  of  the  instrument, 
the  terms  near  rod  and  distant  rod  being  used  to  indicate  the 
relative  position  of  the  rods  for  each  set-up.  Having  taken 
the  four  readings  we  have 

~  _     (sum  of  near-rod  readings)  —  (sum  of  distant-rod  readings) 
(sum  of  distant-rod  intervals)  —  (sum  of  near-rod  intervals) ' 

in  which  C  is  called  the  bubble  error  or  constant  for  the  day's 
work.  If  C  does  not  exceed  0.010  (numerically)  it  is  not  advisable 
to  change  the  adjustment.  The  telescope  looks  down  when  C 
is  positive  and  up  when  C  is  negative,  so  that  if  an  adjustment 
is  found  to  be  necessary  the  line  of  sight  (middle  wire)  is  raised 
or  lowered  on  the  distant  rod  by  C  times  the  corresponding 
interval,  and  the  bubble  tube  adjusted  to  bring  the  bubble 
central.  A  new  determination  of  C  is  always  made  after  each 
adjustment,  and  in  very  precise  work  the  distant-rod  readings 
are  corrected  for  curvature  and  refraction  (Table  VII)  before 
using  in  the  formula,  as  these  errors  double  up  instead  of  canceling 
out  in  this  method  of  adjustment.  A  correction  equal  to  C  times 
the  excess  interval  between  the  foresights  and  backsights  is 
applied  to  the  final  elevation;  if  the  backsights  are  in  excess  the 
correction  has  the  same  sign  as  C,  and  the  opposite  sign  when  the 
foresights  are  in  excess. 

90c.  Use  of  Coast  Survey  Level.  In  order  to  obtain  the  best 
results  with  this  instrument  all  the  precautions  given  in  Art.  88, 
and  briefly  summarized  in  Art.  89c,  must  be  observed.  The 
program  of  observations  is  much  simpler  than  with  the  European 
level,  there  being  nothing  to  do  at  each  set-up  except  to  obtain 
the  three  wire  readings  on  each  rod,  with  the  bubble  kept  exactly 
centered  while  reading  each  wire.  It  is  considered  advisable 
to  read  the  fore  rod  first  on  every  other  set-up.  In  the  precise 
leveling  of  the  U.  S.  Coast  and  Geodetic  Survey  a  correction  for 
excess  of  sights  is  applied  for  curvature  and  refraction  and  also 
for  bubble  error,  together  with  corrections  for  absolute  length 
of  rod  and  average  temperature  of  rod.  An  example  illustrating 
the  keeping  of  the  notes  is  given  on  the  next  page. 


GEODETIC  LEVELING 


157 


^ 


bD 

2 


g 

I 

BrK, 


& 


MH    " 

P 

09 
| 

Sum  o 
Interva 

1C                      t^                      GO                      1O 
O                      OS                      00                      ^ 
•^                      IO                      1>                      GO 

11 

OS  O  OS        O  O  O        OS  OS  OS        OS  OS  OS        CM  CM  *O 

0 

1—1 

CM 

H'l 

CO                    O                    O                    IT-                    t^        l>  GO 

El 

Jj 

CN                    00                    CO                    Th                       .'        ira  ^_i 

OQ  73 

1 

CO                     I-H                     O                     lO                     CO         OS  CO 
t~»                         1O                          CO                          OS                          O          OO  OS 

CM                  CM                  CM                  CM                  I-H        T—  i  r~ 

1S 

~    1 

'C    § 

_,  M-g 

c  *p3 

|.§3 

S'O'S 
£«2 
Hl|| 

III    111    III    III    III 

K 

|d 

^1 

|1 

>c%           ^co           >n           f^co           >c% 

o  a° 

rtH 

S-& 

IB 

*gJ3 

t:  " 

S  & 

00                    lO                    OS                    CM 

l] 

W 

"SI 
s  5 

o,»oo      <^0      000=^      £SS      «Sg 

55  § 

H| 

i—  1           i-H  i—  1  CM                            i-H                            i—  1 

SH  M 

&  a 

a  a 

O                    CO                    CO                    O                    CO        OS 

2  CO 

d 

CO                      0                      CM                      g                      CO         CO 

c  >> 

O                    f^                    O                    O                    ^^        CO 

_c8  ^o 

'o.   <£> 

||| 

^COd         lO  i—  I  »O         T^rM'—  i         QO»OC<I         I>-COO 
t^»  F*"  tT^*         G^  CO  CO          OO  00  00          Ot*  tt>  Oi          C^  lO  00 

IB 

I" 

« 

N' 

o  ^s 

3            3            5            S            ^ 

^^  ; 

1 

158  GEODETIC  SURVEYING 

91.  Rods  and  Turning  Points.  Various  types  of  rods  and 
turning  points  have  been  used  in  precise  level  work,  with  details 
changing  from  time  to  time.  The  notes  here  given  are  intended 
to  briefly  cover  the  points  of  interest  to  engineers. 

Rods.  Precise  leveling  rods  are  now  generally  made  of  wood, 
sometimes  soaked  in  melted  paraffin  to  eliminate  changes  of  length 
by  absorption  of  atmospheric  moisture,  cross  or  T-shaped  in 
section,  about  3.5  meters  in  length,  graduated  metrically,  pro- 
vided with  a  plumb  line  or  level,  and  designed  to  be  used  with- 
out targets.  The  Coast  Survey  rod  is  cross-shaped  in  section, 
of  pine  wood  which  has  absorbed  about  20  per  cent  of  its  original 
weight  of  paraffin,  graduated  to  centimeters  and  read  by  estima- 
tion to  millimeters,  and  provided  with  a  circular  level  for  making 
it  vertical.  Target  rods  were  abandoned  by  the  Coast  Survey 
in  1899.  For  a  description  of  Coast  Survey  rods  see  Appendix 
No.  8,  Report  for  1895,  and  Appendix  No.  8,  Report  for  1900.  The 
precise  rods  used  by  the  Corps  of  Engineers,  U.  S.  A.,  are  similar 
to  the  above,  but  T-shaped  in  cross-section.  The  Molitor 
rod  (designed  by  Mr.  David  S.  Molitor,  and  described  in  Trans. 
Am.  Soc.  C.E.,  Vol.  XLV,  page  12)  is  illustrated  in  Fig.  45,  and 
is  a  precise  rod  of  the  highest  class.  The  smallest  divisions  are 
two  millimeters  wide,  and  the  reading  is  taken  to  millimeters  or 
closer  by  estimation. 

Rod  constant  and  adjustment.  The  precise  leveling  rod  has  one 
constant,  and  one  adjustment.  The  rod  constant  is  its  absolute 
length  between  extreme  divisions,  which  may  differ  slightly 
from  its  designated  length,  and  which  should  be  examined  at 
least  once  a  year.  If  the  rod  is  long  or  short  a  self-evident 
correction  is  required,  which  only  affects  the  final  difference 
of  elevation  between  two  points.  The  rod  adjustment  is  the 
adjustment  of  its  level,  which  should  be  examined  daily  by 
making  the  rod  vertical  with  a  plumb  line,  and  corrected  if 
necessary. 

Turning  points.  Both  foot-plates  and  foot-pins  have  been 
used  for  turning  points.  Cast  iron  foot-plates  about  six  inches 
in  diameter  have  been  used  extensively  by  the  Coast  Survey, 
but  were  practically  abandoned  in  1903  as  inferior  to  pins.  Fig.  45 
shows  a  style  of  foot-pin  first  used  by  Prof.  J.  B.  Johnson  in  1881, 
and  meeting  every  requirement  of  a  good  pin.  It  is  driven  nearly 
flush  with  the  ground  with  a  wooden  mallet.  Such  a  pin  is 


GEODETIC  LEVELING 


159 


•I 


Spirit 
level. 

Handles. 


I 


FIG.  45. — Molitor's  Precise-level  Rod  and  Johnson's  Foot-pin. 


160  GEODETIC  SURVEYING 

best  made  of  steel.  The  little  groove  in  the  head  is  to  prevent 
dust  or  sand  from  settling  on  the  bearing  point. 

92.  Adjustment  of  Level  Work.  In  running  level  lines  of 
any  importance  the  work  is  always  arranged  so  as  to  furnish 
a  check  on  itself,  or  to  connect  with  other  systems,  and  a  cor- 
responding adjustment  is  required  to  eliminate  the  discrepancies 
which  appear.  The  problem  may  always  be  solved  by  the  method 
of  least  squares  when  definite  weights  have  been  assigned  to  the 
various  lines.  When  the  work  is  all  of  the  same  grade  the  lines 
are  weighted  inversely  as  their  length.  This  rule  requires  an 
error  to  be  distributed  uniformly  along  any  given  line  to  adjust 
the  intermediate  points.  A  common  rule  for  intermediate  points 
on  a  line  or  circuit  is  to  distribute  the  error  as  the  square  root 
of  the  various  lengths;  but  as  this  rule  is  inconsistent  with  itself 
it  is  not  recommended.  The  following  rules  for  the  adjustment 
of  level  work  will  usually  be  found  sufficient  and  satisfactory. 

Duplicate  lines.  A  duplicate  line  is  understood  to  mean  a 
line  run  over  the  same  route,  but  in  the  opposite  direction  and 
with  different  turning  points.  This  is  the  best  way  of  checking 
a  single  line  of  levels.  The  discrepancy  which  usually  appears 
is  divided  equally  between  the  two  lines. 

Simultaneous  lines.  These  are  lines  run  over  the  same  route 
in  the  same  direction,  but  with  different  turning  points.  In 
this  case  the  final  elevation  is  taken  as  the  mean  of  the  elevations 
given  by  the  different  lines. 

Multiple  lines.  This  is  understood  to  mean  two  or  more 
lines  run  between  two  points  by  different  routes.  In  this  case 
the  difference  of  elevation  as  given  by  each  line  is  weighted  inversely 
as  the  length  of  that  line,  and  the  weighted  arithmetic  mean 
is  taken  as  the  most  probable  difference  of  elevation.  Thus  if 
the  difference  of  elevation  between  A  and  B  is  9.811  by  a  6-mile 
line,  9.802  by  an  8-mile  line,  and  9.840  by  a  12-mile  line,  we  have 

Mean  difference  of  elevation 

^  (9.811  X  j)  +  (9.802  X  |)  +  (9.840  X  A)      Q  S11 

Intermediate  points.  These  may  occur  on  a  line  whose  ends 
have  been  satisfactorily  adjusted  or  on  a  closed  circuit.  In 
either  case  the  required  adjustment  is  distributed  uniformly 
throughout  the  line,  making  the  correction  between  any  two 


GEODETIC  LEVELING 


161 


points  directly  proportional  4to  the  length  between  those  two 
points. 

Level  nets.  Any  combination  of  level  lines  forming  a  series 
of  closed  circuits  is  called  a  polygonal  system  or  level  net.  Fig.  46 
represents  such  a  system.  If  the  true  difference  of  elevation 
were  known  from  point  to  point,  then  the  algebraic  sum  of  the 
differences  in  any  closed  circuit  would  always  equal  zero,  the 
rise  and  fall  balancing.  In  practical  work  the  various  circuits 
seldom  add  up  to  zero,  and  an  adjustment  has  to  be  made  to 
eliminate  the  discrepancies.  A  rigor- 
ous adjustment  requires  the  use  of 
the  method  of  least  squares,  but  the 
approximate  adjustment  here  described 
will  generally  give  very  nearly  the  same 
results.  Pick  out  the  circuit  which 
shows  the  largest  discrepancy,  and 
distribute  the  error  among  the  differ- 
ent lines  in  direct  proportion  to  their 
length.  Take  the  circuit  showing  the 
next  largest  discrepancy,  and  distribute 
its  error  uniformly  among  any  of  its 
lines  not  previously  adjusted  in  some 
other  circuit,  continuing  in  this  way 
until  all  the  circuits  have  been  ad- 
justed. The  circuits  here  intended  are 

the  single  closed  figures,  as  BEFC,  and  not  such  a  circuit  as 
ABEFCA;  and  no  attention  is  to  be  paid  to  the  direction  or 
combination  in  which  the  lines  may  have  been  run. 

93.  Accuracy  of  Precise  Spirit  Leveling.  The  accuracy 
attainable  in  precise  spirit  leveling  may  be  judged  by  noting  the 
discrepancies  between  duplicate  lines  (Art.  92).  On  the  U.  S. 
Coast  and  Geodetic  Survey  the  limit  of  discrepancy  allowed 
between  duplicate  lines  is  4mm.  ^/K,  meaning  4  millimeters 
multiplied  by  the  square  root  of  the  distance  in  kilometers  between 
the  ends  of  the  lines;  if  this  limit  is  exceeded  the  line  must  be 
rerun  both  ways  until  two  results  are  obtained  which  fall  within 
the  specified  limits.  In  various  important  surveys  the  allowable 
limit  has  ranged  from  5mm.  \/K  to  10mm. \/K,  or  0.021ft. \/M 
to  0.042ft.  VM  where  M  is  the  distance  in  miles.  The  probable 
error  of  the  mean  result  of  a  pair  of  duplicate  lines  is  practically 


162  GEODETIC  SURVEYING 

one-third  of  the  discrepancy,  and  in  actual  work  of  the  highest 
grade  falls  below  Imm.v'jK'.  The  adjusted  value  of  the  eleva- 
tion above  mean  sea  level  of  Coast  Survey  bench  mark  K  in 
St.  Louis  has  a  probable  error  of  only  32  millimeters  or  about 
li  inches,  and  it  is  almost  certain  that  no  amount  of  leveling 
will  ever  change  the  adopted  elevation  as  much  as  6  inches. 

A  much  more  severe  test  of  the  accuracy  of  leveling  is  obtained 
from  the  closures  of  large  circuits  running  up  sometimes  to  1000 
or  more  miles  in  circumference.  The  greatest  error  indicated 
by  the  circuit  closures  in  any  line  in  about  20,000  miles  of  precise 
spirit  leveling  executed  by  the  U.  S.  Coast  and  Geodetic  Survey 
and  other  organizations,  is  about  one-tenth  of  an  inch  per  mile. 
With  the  Coast  Survey  level  of  Art.  90  very  much  closer  results 
have  been  reached. 


CHAPTER  VII 

ASTRONOMICAL  DETERMINATIONS 

94.  General  Considerations.  The  astronomical  determina- 
tions required  in  practical  geodesy  are  Time,  Latitude,  Longitude 
and  Azimuth.  The  precise  determination  of  these  quantities 
requires  special  instruments  as  well  as  special  knowledge  and  skill, 
and  falls  within  the  province  of  the  astronomer  or  professional 
geodesist  rather  than  that  of  the  civil  engineer.  A  fair  deter- 
mination, however,  of  one  or  more  of  these  quantities  is  not 
infrequently  required  of  the  engineer,  so  that  a  partial  knowledge 
of  the  subject  is  necessary.  A  complete  discussion  of  the  sub- 
jects of  this  chapter  may  be  found  in  Doolittle's  Practical  Astron- 
omy, or  in  Appendix  No.  7,  Report  for  1897-98,  U.  S.  Coast  and 
Geodetic  Survey.  As  the  work  of  the  fixed  observatory  is  out- 
side the  sphere  of  the  engineer,  the  following  articles  are  intended 
to  cover  field  methods  only. 

The  instruments  used  by  the  engineer  will  generally  be  limited 
to  the  sextant,  the  engineer's  transit,  one  of  the  higher  grades 
of  transits,  or  the  altazimuth  instruments  of  Chapter  III.  All 
of  these  instruments  are  suitable  for  either  day  or  night  observa- 
tions, except  that  the  ordinary  engineer's  transit  is  not  usually 
furnished  with  means  for  illuminating  the  cross-hairs  at  night. 
This  difficulty  may  be  overcome  by  substituting  in  place  of  the 
sunshade  a  similar  shade  of  thin  white  paper,  a  flat  piece  of  bright 
tin  bent  over  in  front  of  the  object  glass  at  an  angle  of  about  45° 
and  containing  an  oblong  hole  having  a  slightly  less  area  than 
that  of  the  lens,  or  a  special  reflecting  shade  which  may  be  bought 
from  the  maker  of  the  instrument.  The  light  of  a  bull's-eye 
lantern  thrown  on  any  of  these  devices  will  render  the  cross-hairs 
visible. 

In  astronomical  work  the  observer  is  assumed  to  be  at  the 
center  of  the  earth,  this  point  being  taken  as  the  center  of  a  great 

163 


164  GEODETIC  SURVEYING 

celestial  sphere  on  which  all  the  heavenly  bodies  are  regarded  as 
being  projected.  Any  appreciable  errors  arising  from  the  assumption 
that  the  earth  is  stationary  or  that  the  observer  is  at  its  center, 
are  duly  corrected.  All  vertical  and  horizontal  planes  and  the 
planes  of  the  earth's  equator  and  meridians  are  imagined  extended 
to  an  intersection  with  the  celestial  sphere,  and  are  correspond- 
ingly named.  Fig.  47,  page  166,  is  a  diagram  of  the  celestial 
sphere,  and  the  accompanying  text  contains  the  definitions  and 
notation  used  in  the  discussions.  A  thorough  study  and  compre- 
hension of  the  figure  and  text  are  absolutely  essential  for  an 
understanding  of  what  follows.  The  necessary  values  of  the 
right  ascensions,  declinations,  etc.,  required  in  the  formulas,  are 
obtained  from  the  American  Ephemeris,  commonly  called  the 
Nautical  Almanac,  which  is  issued  yearly  (three  years  in  advance) 
by  the  Government. 


TIME 


95.  General  Principles.    Time  is  measured  by  the  rotation 
of  the  earth  on  its  axis,  which  may  be  considered  perfectly  uniform 
for  the  closest  work.     The  rotation  is  marked  by  the  observer's 
meridian   sweeping   around   the   heavens.     The   intersection   of 
this  meridian  with  the  celestial  equator  furnishes  a  point  whose 
uniform  movement  around  the  equator  marks  off  time  in  angular 
value.     The  angle  thus  measured  at  any  moment  between  the 
observer's  meridian  and  the  meridian  of  any  given  point  (which 
may  itself  be  moving)  is  the  hour  angle  of  that  point  at  that 
moment.     These  angles  are,  of  course,  identical  with  the  cor- 
responding spherical  angles  at  the  pole.     When  360°  of  the  equa- 
tor have  passed  by  the  meridian  of  a  reference  point  (whether 
moving  or  not)  the  elapsed  time  is  called  twenty-four  hours,  so 
that  any  kind  of  time  is  changed  from  angular  value  to  the  hour 
system  by  dividing  by  15,  and  vice  versa.     There  are  two  kinds 
of  time  in  common  use,  mean  solar  time  and  sidereal  time,  based 
on  the  character  of  the  reference  point.     Mean  solar  time  is  the 
ordinary  time  of  civil  life,  and  sidereal  time  is  the  time  chiefly  used 
in  astronomical  work. 

96.  Mean  Solar  Time.    The  fundamental  idea  of  solar  time  is  to 
use  as  the  measure  of  tune  the  apparent  daily  motion  of  the  sun 


ASTRONOMICAL  DETERMINATIONS  165 

around  the  earth;  this  is  called  apparent  solar  time,  the  upper  transit 
of  the  sun  at  the  observer's  meridian  being  called  apparent  noon. 
Apparent  solar  time,  however,  is  not  uniform,  on  account  of  a 
lack  of  uniformity  in  the  apparent  annual  motion  of  the  sun 
around  the  earth.  This  is  due  to  the  fact  that  the  apparent 
annual  motion  is  in  the  ecliptic,  the  plane  of  which  makes  an  angle 
with  the  plane  of  the  equator,  and  the  further  fact  that  even  in 
the  ecliptic  the  apparent  motion  is  not  uniform.  To  overcome 
this  difficulty,  a  fictitious  sun,  called  the  mean  sun,  is  assumed  to 
move  annually  around  the  equator  at  a  perfectly  uniform  rate, 
and  to  make  the  circuit  of  the  equator  in  the  same  total  time  that 
the  true  sun  apparently  makes  the  circuit  of  the  ecliptic.  Mean 
solar  time  is  time  as  indicated  by  the  apparent  daily  motion  of 
the  mean  sun  and  is  perfectly  uniform.  The  difference  between 
apparent  solar  time  and  mean  solar  time  is  called  the  equation 
of  time,  varies  both  ways  from  zero  to  about  seventeen  minutes, 
and  is  given  in  the  Nautical  Almanac  for  each  day  of  the  year. 
Local  mean  time  for  any  meridian  is  the  hour  angle  of  the  mean 
sun  measured  westward  from  that  meridian,  local  mean  noon 
being  the  time  of  the  upper  transit  of  the  mean  sun  for  that 
meridian. 

96a.  Standard  Time.  This  time,  as  now  used  in  the  United 
States,  is  mean  solar  time  for  certain  specified  meridians,  each 
district  using  the  time  of  one  of  these  standard  meridians  instead 
of  its  own  local  time.  The  meridians  used  are  the  75th,  90th, 
105th  and  120th  west  of  Greenwich,  furnishing  respectively 
Eastern,  Central,  Mountain  and  Pacific  standard  time.  Standard 
time  for  all  points  in  the  United  States  differs  only  by  even  hours, 
with  very  large  belts  having  exactly  the  same  time,  the  variation 
from  local  mean  time  seldom  exceeding  a  half  hour.  In  the  lat- 
itude of  New  York  local  mean  time  varies  about  four  seconds 
for  every  mile  east  or  west.  Standard  time  may  be  obtained  at 
any  telegraph  station  with  a  probable  error  of  less  than  a  second. 
In  all  astronomical  work  standard  time  must  be  changed  to  local 
mean  time. 

96b.  To  Change  Standard  Time  to  Local  Mean  Time  and  vice 
versa.  The  difference  between  standard  time  and  local  mean 
time  at  any  point  equals  the  difference  of  longitude  (expressed 
in  time  units,  Art.  113)  between  the  given  point  and  the  standard 
time  meridian  used.  For  points  east  of  the  standard  time 


166 


GEODETIC  SURVEYING 

z 


North 
Poleo 


FIG.  47. — The  Celestial  Sphere. 

EXPLANATION 

HZH'N  =  meridian  of  observer; 
Z,  W,  N  =  points  on  prime  vertical; 
M,  m  =  projection  of  azimuth  marks  on  celestial  sphere; 
Z=observer's  zenith; 
N  =  observer's  nadir; 

Angles  at  Z,  and  corresponding  horizontal  angles  at  0,  are  azimuth  angles; 
Angels  at  P,  and  corresponding  equatorial  angles  at  0,  are  hour  angles. 

CONVERSION  OF  ARC  AND  TIME 


Arc.  Time. 

1  °  =  4  minutes 
1'  =4  seconds 
1"  =  is  second 


Time.  Arc. 

1  hour  =  15° 

1  minute  =  15' 

1  second  =  15" 


ASTRONOMICAL  DETERMINATIONS  167 


DEFINITIONS 

The  zenith  (at  a  given  station)  is  the  intersection  of  a  vertical  line  with 
the  upper  portion  of  the  celestial  sphere. 

The  nadir  is  the  intersection  of  a  vertical  line  with  the  lower  portion 
of  the  celestial  sphere. 

The  meridian  plane  is  the  vertical  plane  through  the  zenith  and  the  celes- 
tial poles,  the  meridian  being  the  intersection  of  this  plane  with  the  celestial 
sphere. 

The  prime  vertical  is  the  vertical  plane  (at  the  point  of  observation)  at 
right  angles  with  the  meridian  plane. 

The  latitude  of  a  station  is  the  angular  distance  of  the  zenith  from  the 
equator,  and  has  the  same  value  as  the  altitude  of  the  elevated  pole.  Lati- 
tude may  also  be  defined  as  the  declination  of  the  zenith.  North  latitude 
is  positive  and  south  latitude  negative. 

Co-latitude  =  90°  -  latitude. 

Right  ascension  is  the  equatorial  angular  distance  of  a  heavenly  body 
measured  eastward  from  the  vernal  equinox. 

Declination  is  the  angular  distance  of  a  heavenly  body  from  the  equator. 
North  declination  is  positive  and  south  declination  negative. 

Co-declination  or  polar  distance  =90°—  declination. 

The  hour  angle  of  a  heavenly  body  is  its  equatorial  angular  distance 
from  the  meridian.  Hour  angles  measured  towards  the  west  are  positive, 
and  vice  versa. 

The  azimuth  of  a  heavenly  body  (or  other  point)  is  its  horizontal  angular 
distance  from  the  south  point  of  the  meridian  (unless  specified  as  from  the 
north  point).  Azimuth  is  positive  when  measured  clockwise,  and  vice 
versa. 

The  altitude  of  a  heavenly  body  is  its  angular  distance  above  the  horizon. 

Co-altitude  or  zenith  distance  =  90°—  altitude. 

Refraction  is  the  angular  increase  in  the  apparent  elevation  of  a  heavenly 
body  due  to  the  refraction  of  light,  and  is  always  a  negative  correction. 

Parallax  (in  altitude)  is  the  angular  decrease  in  the  apparent  elevation 
of  a  heavenly  body  due  to  the  observation  being  taken  at  the  surface  instead 
of  at  the  center  of  the  earth,  and  is  always  a  positive  correction. 

NOTATION 

<£  =  latitude  (+  when  north,  —  when  south); 
a  =  right  ascension; 

5  =  declination  (+  when  north,  —  when  south); 
<=hour  angle  (+  to  west,  —  to  east); 

A  =  azimuth  from  north  point  (+  when  measured  clockwise); 
Z  =  azimuth  from  south  point  (+  when  measured  clockwise); 
h  =  altitude; 
z  =  zenith  distance; 
r  =  refraction; 
p=  parallax. 


168  GEODETIC  SURVEYING 

meridian  local  mean  time  is  later  than  standard  time,  and  vice 
versa. 

Example  1.  New  York,  N.Y.,  uses  75th-meridian  standard  time.  Given 
the  longitude  of  Columbia  College  as  73°  58'  24".6  west  of  Greenwich,  what 
is  the  local  mean  time  at  10h  14m  17s. 2  P.M.  standard  time? 

75°  00'  00".0  10h  14m  178.2  P.M. 

73     58    24  .6  4      06  .4 


15)  1°  01'  35".4         Ans.  =10h  18m  238.6  P.m. 
4m06s.4 


Example  2.  Philadelphia,  Pa.,  uses  75th-meridian  standard  time.  Given 
the  longitude  of  Flower  Observatory  as  5h  Olm  068.6  west  of  Greenwich,  what 
is  the  standard  time  at  9h  06m  18s.  1  A.M.  local  mean  time. 

15)75°  00'  00".Q  9h  06m  18s.  1  A.M. 


5h  oom  OQS.O  1      06  .6 

5    01    06  .6  - 


Ans.  =9h  07m  24S.7  A.M. 


Im068.6 


97.  Sidereal  Time.  In  this  kind  of  time  a  sidereal  day  of 
twenty-four  hours  corresponds  exactly  to  one  revolution  of  the 
earth  on  its  axis,  as  marked  by  two  successive  upper  transits 
of  any  star  over  the  same  meridian.  The  sidereal  day  for  any 
meridian  commences  when  that  meridian  crosses  the  vernal 
equinox,  and  runs  from  zero  to  twenty-four  hours.  The  sidereal 
time  at  any  moment  is  the  hour  angle  of  the  vernal  equinox  at 
that  moment,  counting  westward  from  the  meridian.  As  the 
right  ascensions  of  stars  and  meridians  are  counted  eastward 
from  the  vernal  equinox,  it  follows  that  the  sidereal  time 
for  any  observer  is  the  same  as  the  right  ascension  of  his 
meridian  at  that  moment.  Hence  when  a  star  of  known 
right  ascension  crosses  the  meridian  the  sidereal  time 
becomes  known  at  that  moment.  The  right  ascension 
of  the  mean  sun  at  Greenwich  mean  noon  (called  sidereal 
time  of  Greenwich  mean  noon)  is  given  in  the  Nautical 
Almanac  for  every  day  of  the  year,  and  is  readily  found 
for  local  mean  noon  at  any  other  meridian  by  adding  the 
product  of  9.8565  seconds  by  the  given  longitude  west  of  Green- 
wich expressed  in  hours. 


ASTRONOMICAL  DETERMINATIONS  169 

98.  To  Change  a  Sidereal  to,  a  Mean  Time  Interval,  and  vice 
versa.    Owing  to  the  relative  directions  in  which  the  earth  rotates 
on  its  axis  and  revolves  around  the  sun  the  number  of  sidereal 
days   in  a  tropical  year  (one  complete  revolution  of  the  earth 
around  the  sun)  is  exactly  one  more  than  the  number  of  solar 
days.     According  to  Bessel  the  tropical  year  contains  365.24222 
mean  solar  days,  hence  365.24222  mean  solar  days  =  366.24222 
sidereal  days,  and  therefore 

1  mean  solar  day=  1.0027379  sidereal  days; 
1  sidereal  day       =  0.9972696  mean  solar  days; 

whence  if  Is  is  any  sidereal  interval  of  time  and  Im  the  mean  solar 
interval  of  equal  value,  we  have 

Is  =Im+  0.0027379  /„      (log  0.0027379  =  7.4374176  -  10) 
Im=I8   ~  0.0027304  7.       (log 0.0027304  =  7.4362263  -  10) 

Where  there  is  much  of  this  work  to  be  done  the  labor  of  computa- 
tion is  lessened  by  using  the  tables  found  in  the  Nautical  Almanac 
and  books  of  logarithms. 

99.  To  Change  Local  Mean  Time  or  Standard  Time  to  Sidereal. 
For  local  mean  time  this  is  done  by  converting  the  mean  time 
interval  between  the  given  time  and  noon  into  the  equivalent 
sidereal  interval  (Art.  98),  and  combining  the  result  with  the 
sidereal  time  of  mean  noon  for  the  given  place  and  date.  Since 
the  right  ascension  of  the  mean  sun  increases  360°  or  twenty- 
four  hours  in  one  year,  the  increase  per  day  will  be  3m  56S.555, 
or  9S.8565  per  hour.     The  sidereal  time  of  mean  noon  for  the 
given  place  is  therefore  found  by  taking  the  sidereal  time  of  Green- 
wich mean  noon  from  the  Nautical  Almanac  and  adding  thereto 
the  product  of  9S.8565  by  the  longitude  in  hours  of  the   given 
meridian,   counted  westward  from  the  meridian  of  Greenwich. 
If  standard  time  is  used  it  must  first  be  changed  to  local  mean 
time  (Art.  966)  before  applying  the  above  rule. 

Example.  To  find  the  sidereal  time  at  Syracuse,  N.  Y.,  longitude 
76°  08'  20". 40  west  of  Greenwich,  when  the  standard  (75th  meridian)  time 
is  10h  42m  00s  AM.,  January  17th,  1911. 


170  GEODETIC  SUEVEYING 

76°  08'  20". 40  10h  42mOOs.OO  standard  time 

75  -  4    33  .36 


15)  1°  08'  20". 40  10    37    26 . 64  local  mean  time 

4m33«>.36  12 


log 4953. 36     =3.6948999  lh  22m 33s. 36  =  4953s. 36 

log 0.0027379  =  7. 4374176  +13   .56 


log  (13s. 56)     =1.1323175  1    22    46  . 92  sidereal  interval 


15)76°  08'  20"  40  log  9.8565  =  0.9937227 

15)76    08    20    .40  log  5.0759=0.7055131 

5h  04m  338 . 36  & 

=  5 .0759  hrs.  log  (50S.03)  =  1 . 6992358 


Sidereal  time  of  Greenwich  mean  noon      19h  43m  09s. 48 
Reduction  to  Syracuse  meridian  4-  50  .03 


Sidereal  time  of  Syracuse  mean  noon         19    43    59.51 
Sid.  int.  from  Syracuse  mean  noon       -1    22    46.92 


Sidereal  time  at  given  instant  18h  21m  12s.  59 

100.  To  Change  Sidereal  to  Local  Mean  Time  or  Standard 
Time.  This  is  the  reverse  of  the  process  in  Art.  99,  and  consists 
in  finding  the  difference  between  the  given  time  and  the  sidereal 
time  of  mean  noon  for  the  given  place  and  date,  changing  this 
interval  to  the  corresponding  mean  time  interval  (Art.  98),  and 
combining  the  result  with  twelve  o'clock  (mean  noon)  by  addi- 
tion or  subtraction  as  the  case  requires.  The  result  is  local  mean 
time,  and  if  standard  time  is  wanted  it  is  then  obtained  as 
explained  in  Art.  966. 

Example.  To  find  the  local  mean  time  and  standard  (75th  meridian) 
time  at  Syracuse,  N.  Y.,  longitude  76°  08'  20".40  west  of  Greenwich,  when 
the  sidereal  time  it  18h  21m  128.59,  January  17,  1911. 

76°  08'  20". 40- 75°  =  1°  08'  20". 40  =  4m  33s. 36 
log  9. 8565  =0.9937227 

15)76°  08'  20".4Q  5.0759=0.7055131 

5h  04m 33s. 36 
=  5 . 0759  hrs.  log  (50* .  03)  r.  1 . 6992358 


ASTRONOMICAL  DETERMINATIONS  171 


Sidereal  time  of  Greenwich  mean  noon  19h  43m  09s. 48 

Reduction  to  Syracuse  meridian  +  50  .03 


Sidereal  time  of  Syracuse  mean  noon  19    43    59.51 

Sidereal  time  at  given  instant  18    21     12.59 


Sidereal  interval  before  Syracuse  mean  noon        lh  22m  46s.  92 


lh  22m  46s.  92  =  4966s.  92 

log  4966  .92     -3.  6960872     Reduction  to  ,       1*  22™  46s.  92 
log  0.0027304  =  7.  4362263     mean  time  -   13  .56 


_ 

log  (13s.  56)     =1.1323135  I  interval  1     22    33    36 

12 


Local  mean  time  at  given  instant  (morning)       10h  37m  26s. 64 
Reduction  to  standard  time  +4    33  . 36 


Standard  time  at  given  instant  (morning)  10h  42m  00s. 00 

101.  Time  by  Single  Altitudes.  The  altitude  of  any  heavenly 
body  as  seen  by  an  observer  at  a  given  point  is  constantly  chang- 
ing, each  different  altitude  corresponding  to  a  particular  instant 
of  time  which  can  be  computed  if  the  latitude  and  longitude  are 
approximately  known.  In  finding  local  mean  time  or  sidereal 
time  it  is  sufficient  to  know  the  latitude  to  the  nearest  minute 
and  the  longitude  within  a  few  degrees.  In  changing  from  local 
to  standard  time,  however,  an  error  of  1s  will  be  caused  by  each 
15"  error  of  longitude.  If  the  latitude  is  not  known  it  may 
generally  be  scaled  sufficiently  close  from  a  good  map,  or  it  may 
be  determined  as  explained  in  Arts.  107  or  108.  By  comparing 
the  observed  time  for  a  certain  measured  altitude  of  sun  or  star 
with  the  corresponding  computed  time  the  error  of  the  observer's 
timepiece  is  at  once  determined.  The  observation  may  be 
made  with  a  transit  (or  altazimuth  instrument),  or  with  a  sextant 
(and  artificial  horizon),  the  latter  being  the  most  accurate.  In 
either  case  several  observations  ought  to  be  taken  in  imme- 
diate succession,  as  described  below,  and  the  average  time  and 
average  altitude  used  in  the  reductions.  The  probable  error  of 
the  result  may  be  several  seconds  with  a  transit,  and  a  second 
or  two  with  the  sextant.  The  actual  error  is  apt  to  be  larger  on 
account  of  the  uncertainties  of  refraction.  The  observation  is 
commonly  made  with  the  sextant  and  on  the  sun. 


172  GEODETIC  SURVEYING 

lOla.  Making  the  Observation.  The  best  time  for  making 
an  observation  on  the  sun  is  between  8  and  9  o'clock  in  the 
morning  and  between  3  and  4  o'clock  in  the  afternoon,  in  order 
to  secure  a  rapidly  changing  altitude  and  at  the  same  time  avoid 
irregular  refraction  as  far  as  possible.  The  altitude  of  the 
center  of  the  sun  is  never  directly  measured,  but  the  observations 
are  taken  on  either  the  upper  or  lower  limb,  or  preferably  an  equal 
number  of  times  on  each  limb.  Star  observations  may  be  made 
at  any  hour  of  the  night,  selecting  stars  which  are  about  three 
hours  from  the  meridian  and  near  the  prime  vertical,  and  hence 
changing  rapidly  in  altitude  at  the  time  and  place  of  observation. 
If  two  stars  are  observed  at  about  the  same  time  having  about 
the  same  declination  and  about  the  same  altitude,  but  lying  on 
opposite  sides  of  the  meridian,  the  mean  of  the  two  results  (de- 
terminations of  the  clock  error)  will  be  largely  free  from  the  errors 
due  to  the  uncertainties  of  refraction. 

In  taking  the  observation  an  attendant  notes  the  watch 
time  to  the  nearest  second  at  the  exact  moment  the  pointing 
is  made.  //  the  transit  is  used,  an  equal  number  of  readings 
should  be  taken  with  the  telescope  direct  and  reversed,  the  plate 
bubble  parallel  to  the  telescope  being  brought  exactly  central 
for  each  individual  pointing  in  order  to  eliminate  the  instrumental 
errors  of  adjustment.  If  a  star  or  one  limb  of  the  sun  is  observed 
there  should  be  not  less  than  3  direct  and  3  reversed  readings. 
If  both  limbs  of  the  sun  are  observed  there  should  be  not  less 
than  2  direct  and  2  reversed  readings  on  each  limb,  or  3  direct 
on  one  limb  and  3  reversed  on  the  other  limb.  //  the  sextant  and 
artificial  horizon  are  used,  and  the  pointings  are  made  on  a  star 
or  on  one  limb  of  the  sun,  not  less  than  5  readings  of  the  double 
altitude  should  be  taken;  if  both  limbs  of  the  sun  are  observed, 
not  less  than  3  readings  should  be  taken  for  each  limb.  These 
double  altitudes  are  always  corrected  for  index  error  and  some- 
times for  eccentricity.  It  is  considered  better  not  to  use  the 
cover  on  the  artificial  horizon,  but  if  it  has  to  be  done  it  should 
be  reversed  on  half  of  the  readings.  If  as  much  tin  foil  is  added 
to  commercial  mercury  as  it  will  unite  with,  an  amalgam  is  formed 
whose  surface  is  not  readily  disturbed  by  the  wind,  thus  rendering 
the  cover  unnecessary.  When  the  mercury  is  poured  in  its 
dish  it  must  be  skimmed  with  a  card  to  clean  its  reflecting  surface. 
In  all  of  the  above  methods  of  observing,  the  work  is  supposed 


ASTRONOMICAL  DETERMINATIONS  173 

to  be  carried  on  with  reasonable  regularity  and  expedition  when 
once  started.  With  'any  method  it  is  desirable  to  take  at  least 
two  sets  of  readings  and  compute  them  independently  as  a  check, 
the  extent  of  the  disagreement  showing  the  quality  of  the  work 
that  has  been  done,  while  the  mean  value  is  probably  nearer  the 
truth  than  the  result  of  any  single  set. 

lOlb.  The  Computation.  The  first  step  in  the  computation 
of  any  set  of  observations  is  to  find  the  average  value  of  the  meas- 
ured altitudes  and  the  average  value  of  the  recorded  times,  these 
average  values  constituting  the  observed  altitude  and  time  for 
that  set.  This  observed  altitude  is  then  reduced  to  the  true 
altitude  for  the  center  of  the  object  observed.  The  reductions 
which  may  be  required  are  for  refraction,  parallax,  and  semi- 
diameter.  The  apparent  altitude  of  all  heavenly  bodies  is  too 
large  on  account  of  the  refraction  of  light;  Table  VIII  gives  the 
average  angular  value  of  refraction,  which  is  a  negative  correc- 
tion for  all  measured  altitudes.  Parallax  is  an  apparent  dis- 
placement of  a  heavenly  body  due  to  the  fact  that  the  observer 
is  not  at  the  center  of  the  earth;  star  observations  require  no 
correction  for  parallax;  all  solar  observations  require  a  positive 
correction  for  parallax,  the  amount  being  equal  to  8".9  multiplied 
by  the  cosine  of  the  observed  altitude.  The  correction  for 
semi-diameter  is  only  required  in  solar  work,  and  not  even  then 
for  the  average  of  an  equal  number  of  observations  on  both  limbs; 
when  the  average  altitude  refers  to  only  one  limb  a  self-evident 
positive  or  negative  correction  is  required  for  semi-diameter, 
the  value  of  which  is  given  in  the  Nautical  Almanac  for  the  me- 
ridian of  Greenwich  for  every  day  of  the  year,  and  can  readily 
be  interpolated  for  the  given  longitude.  Letting  h  equal  true 
altitude  for  center,  h'  equal  measured  altitude,  r  equal  refrac- 
tion, p  equal  parallax,  and  s  equal  semi-diameter,  we  have 

h  (for  a  star)  =  In!  —  r; 

h  (sun,  both  limbs)  =  h'  —  r  +  p; 

h  (sun,  one  limb)      =  h'  —  r  +  p  ±  s. 

In  the  polar  triangle  ZPS,  Fig.  47,  page  166,  the  three  sides  are 
known.  ZP,  the  co-latitude,  is  found  by  subtracting  the  observer's 
latitude  from  90°.  PS,  the  polar  distance  or  co-declination,  is 


174  GEODETIC  SURVEYING 

found  by  subtracting  the  declination  of  the  observed  body  from  90°. 
In  the  case  of  the  sun  the  declination  is  constantly  changing  and 
must  be  taken  for  the  given  date  and  hour  (the  time  being  always 
approximately  known).  The  sun's  declination  for  Greenwich 
mean  noon  is  given  in  the  Nautical  Almanac  for  every  day  in  the 
year,  and  can  be  interpolated  for  the  Greenwich  time  of  the  observa- 
tion; the  Greenwich  time  of  the  observation  differs  from  the 
observer's  time  by  the  difference  in  longitude  in  hours,  remember- 
ing that  for  points  west  of  Greenwich  the  clock  time  is  earlier,  and 
vice  versa.  ZS,  the  co-altitude,  is  found  by  subtracting  the 
reduced  altitude  h  from  90°.  Using  the  notation  of  Fig.  47, 
we  have  from  spherical  trigonometry 

cos  z  =  sin  <j>  sin  d  +  cos  <j>  cos  d  cos  t, 
whence 

cos  z  —  sin  d>  sin  d 

cos  t  = j — -L— ~ , 

cos  <p  cos  o 

which  for  logarithmic  computation  is  reduced  to  the  form 

.  /sin  4Tz  +  (<f>  —  d}]  sin  *[«  —  (d>  —  d}] 
tan  ^t  =  \ ^ —     ( ,    ,    »v-. ff T,  _,    *\    • 

The  value  of  t  thus  found  is  the  hour  angle  of  the  observed  body, 
or  angular  distance  from  the  observer's  meridian.  Dividing  t 
by  15  changes  the  angular  value  to  the  corresponding  time  interval. 

For  a  solar  observation  the  time  interval  is  subtracted  from  or 
added  to  12  o'clock  according  as  the  sun  is  east  or  west  of  the 
meridian,  giving  the  apparent  solar  time  of  the  observation. 
This  apparent  time  must  be  reduced  to  mean  time  by  applying 
the  equation  of  time  for  the  given  date  and  hour,  taken  from  the 
Nautical  Almanac  in  the  manner  above  described  for  finding  the 
declination.  The  local  mean  time  of  the  observation  as  thus 
found  may  be  changed  to  standard  time  (Art.  966),  or  sidereal 
time  (Art.  99),  if  so  desired. 

For  a  star  observation  the  time  interval  is  subtracted  from  or 
added  to  the  star's  right  ascension  according  as  the  star  is  east 
or  west  of  the  meridian,  giving  the  sidereal  time  of  the  observation. 
This  may  be  changed  to  local  mean  time  (Art.  100),  and  thence 
to  standard  time  (Art.  966),  if  so  desired. 


ASTRONOMICAL  DETERMINATIONS  175 

EXAMPLE.— TIME   BY  SINGLE   ALTITUDES   OF  THE   SUN 


CHICAGO,  ILL.,  June  1,  1911. 
Latitude  =41°  50'  01".  0  N. 

Longitude  =  87°  36'  42".0  =  5h  50m  26s.  8  =  5.  84  hrs.  W.  of  Greenwich! 
Uses  90th  meridian  (Central  Standard)  time  =  6.  00  hrs.  W.  of  Greenwich. 
Local  tune—  Standard  time  =  6h  00m  00s  .  0  -  5h  50m  26s  .  8  =  9m  33s  .  2  . 

Sun.  h'  Watch,  A.M.  h  and  z. 

47°  00'  8h  46mll8 

Lower  limb  <{  47    20  8    48    06  Par.  =  8".  9  X  cos  48°  20'  =  6".  7 

47    40  8    50    03 

49    00  8    54    45  App.  altitude  =  48°  20'  00".  0 

Upper  limb  \  49    20  8    56    41  Refraction      =  51    .3 

49    40  8    68    38  Parallax  +      6    .7 

6)290°  00'     6)53h  14m  24s  ft  =  48°  19'  15".  4 

Average      48°  20'          8h  52m  24~0  z  *=41°  40'  44"T(5 

Approximate  interval  after  Greenwich  mean  noon 

=  8h  52m24s.0  +  6hrs-12hrs  =  2h  52m  24s.  0  =  2.  87  hours. 

Time.  8  dd       Eq.  of  Time. 

At  Greenwich  mean  noon      +21°  56'  33".6     -f  21".23    2m  318.87 

Reduction  for  2.87  hrs.  +      1  00   .8      -  0   .11  1.04 

At  time  of  observation  +21°  57'  34".4     +21".  12     2m  3C8.83 

Equation  of  time  subtractive  from  apparent  time  (on  given  date). 

0.96X2.87_ot)u  21.23  +  21.12^,,  ig 

0.  363X2.  87  =  ls.  04  21.18X2.87   =60".  8 


<j>-d  =19°52'26".6  <f>  +  d  =     63°  47'  35"  .4 

z+(<f>-d)=61    33   11   .2  z+($  +  d)=    105    28  20   .0 

2-<-<5)  =  21    48  18   .0  2-(<   +  £)=-22   06   50   .8 


sin  (30°  46'  35".6)  sin      (10°  54'  09".Q)     „ 
tan^=\cos(52    44   10   .0)  cos  (-11    03   25   .4)  ~ 

<  =  43°  57'  15".4  =  2h  55m49s.O. 

Local  apparent  noon  12h  00m  00s. 0 

Hour  angle  of  sun  2  55    49  .0 

Apparent  solar  time  9h  04m  lls.O 

Equation  of  time  —       2    30  .8 

Local  mean  time  of  observation  9h  Olm  408.2 

Watch  time  of  observation  8    52    24  .0 

Watch  slow  by  mean  time  9m16s .  2 

Reduction  to  standard  time  9    33  .2 

Watch  fast  by  standard  time  Om  17s.O 


176  GEODETIC  SUKVEYING 

In  either  case  the  error  of  the  observer's  timepiece  (as  deter- 
mined by  any  given  set  of  observations)  is  obtained  by  comparing 
the  observer's  average  time  for  the  given  set  with  the  computed 
true  time  for  the  same  set. 

102.  Time  by  Equal  Altitudes.  In  this  method  the  clock 
time  is  noted  at  which  the  sun  (or  a  star)  has  the  same  altitude 
on  each  side  of  the  meridian,  from  which  the  clock  time  of  meridian 
passage  (upper  or  lower  transit  or  culmination)  is  readily  obtained. 
By  comparing  the  clock  time  with  the  true  time  of  meridian 
passage  the  error  of  the  observer's  clock  is  at  once  made  known. 
The  advantages  of  this  method  over  the  method  of  single  altitudes 
are  as  follows:  the  results  are  in  general  more  reliable;  the  com- 
putation is  simpler,  as  it  does  not  involve  the  solution  of  a  spherical 
triangle;  no  correction  is  required  for  refraction,  parallax,  semi- 
diameter,  nor  instrumental  errors;  the  latitude  need  not  be  known 
at  all  for  star  observations,  and  only  very  approximately  for 
solar  work.  The  observations  may  be  made  with  a  transit  or  a 
sextant  (with  artificial  horizon),  the  latter  being  the  most  accurate. 
In  either  case  several  observations  ought  to  be  taken  in  immediate 
succession,  as  described  below,  and  the  average  time  used  in  the 
reductions.  The  probable  error  of  the  result  should  not  exceed 
about  two  seconds  with  the  transit  nor  about  one  second  with 
the  sextant.  The  actual  error  may  be  greater  on  account  of  the. 
uncertainties  of  refraction.  The  method  evidently  assumes  that 
the  refraction  will  be  the  same  for  each  of  the  equal  altitudes, 
but  on  account  of  the  lapse  of  time  between  the  observations 
this  is  not  necessarily  true.  The  observation  is  commonly  made 
with  the  sextant  and  on  the  sun. 

102a.  Making  the  Observation.  As  with  the  previous 
method,  the  best  time  for  making  an  observation  on  the  sun  is 
between  8  and  9  o'clock  in  the  morning  and  between  3  and  4 
o'clock  in  the  afternoon.  The  observations  may  be  taken  entirely 
on  one  limb  of  the  sun  or  an  equal  number  of  times  on  each  limb. 
The  equal  altitudes  may  be  taken  on  the  morning  and  afternoon 
of  the  same  day,  or  on  the  afternoon  of  one  day  and  the  morning 
of  the  next  day.  For  star  observations  a  star  should  be  selected 
which  will  be  about  three  hours  from  the  meridian  and  near  the 
prime  vertical  at  the  times  of  observation.  Since  the  equal 
altitudes  observed  must  be  within  the  hours  of  darkness,  a  star 
is  required  whose  meridian  passage  occurs  within  about  three 


ASTRONOMICAL  DETERMINATIONS  177 

i 

hours  after  dark  and  three  hqjirs  before  daylight.  The  sidereal 
time  of  meridian  passage  is  always  known,  since  it  is  the  same 
as  the  star's  right  ascension,  and  the  corresponding  values  of 
mean  time  and  standard  time  are  readily  found  by  Arts.  100  and 
966.  The  equal  altitudes  may  be  taken  during  the  same  night,  or 
on  the  morning  and  evening  of  the  same  day. 

In  taking  the  observation  the  attendant  notes  the  watch 
time  to  the  nearest  second  at  the  exact  moment  the  pointing  is 
made.  //  the  transit  is  used  the  telescope  is  not  reversed,  but  the 
plate  bubble  parallel  to  the  telescope  is  brought  exactly  central 
for  each  individual  pointing;  no  corrections  are  made  to  the  result- 
ing reading  for  any  instrumental  errors.  //  the  sextant  and 
artificial  horizon  are  used  no  corrections  are  applied  to  the  result- 
ing double  altitude  as  measured.  There  is  no  great  objection 
to  using  the  cover  of  the  artificial  horizon  in  this  method,  and 
when  used  it  is  not  reversed  (as  in  Art.  lOla);  it  is  necessary, 
however,  to  use  it  in  the  same  position  at  both  periods  of  equal 
altitudes. 

If  a  star  or  one  limb  of  the  sun  is  observed  there  should  be 
not  less  than  5  readings  taken  at  each  period  of  equal  altitudes. 
If  both  limbs  of  the  sun  are  observed  there  should  be  not  less  than 
3  readings  (at  each  period)  for  each  limb.  The  angular  readings 
in  this  method  are  always  equally  spaced,  the  instrument  being 
set  in  turn  for  each  equal  change  of  altitude  and  the  time  noted 
when  the  event  occurs.  In  commencing  operations  the  observer 
measures  the  approximate  altitude,  sets  his  vernier  to  the  next 
convenient  even  reading,  and  watches  for  that  altitude  to  be 
reached;  the  next  setting  is  then  made  and  that  altitude  waited 
for,  and  so  on.  At  the  second  period  the  same  settings  must  be 
used,  but  in  reverse  order.  The  size  of  the  angular  interval 
will  depend  on  the  ability  of  the  observer  to  make  each  setting 
in  time  to  catch  the  given  occurrence,  and  can  best  be  found  by 
trial;  under  average  conditions  a  good  observer  would  not  find 
it  difficult  to  use  10'  settings  on  the  transit  and  20'  on  the  sextant. 
It  is  desirable  to  take  at  least  two  independent  sets  of  observa- 
tions, and  compute  them  separately  as  a  check  and  as  an  indica- 
tion of  the  reliability  of  the  results;  the  adopted  value  would 
then  be  taken  as  the  mean  of  the  several  determinations. 

102b.  The  Computation.  In  this  method  there  is  no  object 
in  finding  the  average  of  the  observed  altitudes,  the  method 


178  GEODETIC  SUKVEYING 

being  based  on  the  equality  of  the  corresponding  altitudes  in- 
stead of  their  value.  For  each  set  of  observations,  however, 
it  is  necessary  to  find  the  average  of  the  time  readings  for 
each  of  the  two  periods  of  equal  altitudes.  From  these  values 
the  middle  time  (half-way  point  between  the  two  average  time 
readings)  is  found  for  star  observations,  and  the  middle  time  and 
elapsed  time  (interval  between  average  time  readings)  for  solar 
observations.  For  star  observations  the  middle  time  is  the 
observer's  time  of  meridian  passage.  For  solar  observations 
a  correction  must  be  applied  to  the  middle  time  to  obtain  the 
observer's  time  of  meridian  passage,  on  account  of  the  changing 
declination  of  the  sun. 

For  solar  observations  on  the  same  day,  expressed  in  mean  time 
units,  we  have  from  astronomy 

77  —  M  —  dd_^t/tan  ^       tan  d 

"T5\15iT  "tan 

in  which 

U  =  observer's  time  at  apparent  noon  (upper  transit  of  sun)  ; 
M  =  middle  time  of  the  observations; 
t  =  one-half  elapsed  time,  in  hours  to  three  places    outside 

of  parentheses  and  angular  value  inside  of  parentheses; 
(j>  =  observer's   latitude    (approximate),  +  for  north  and  - 

for  south  latitude; 

§  =  sun 's  declination  at  mean  noon  for  given  date  and  longi- 
tude, +  for  north  and  —  for  south  declination; 
d$  =  hourly  change  of  declination   at  mean   noon   for   given 
date  and  longitude,  +  when  north  declination  is  increasing 
or  south  declination  decreasing,  and  —  when  north  declina- 
tion is  decreasing  or  south  declination  increasing. 

The  values  for  d  and  dd  for  the  given  date  are  found  in  the 
Nautical  Almanac  for  Greenwich  mean  noon  and  interpolated  for 
the  given  meridian.  If  a  sidereal  chronometer  is  used  it  is  neces- 
sary to  convert  t  into  a  mean  time  interval  before  inserting  in  the 
corrective  term  in  the  above  formula,  and  the  value  of  this  term 
must  then  be  reduced  to  a  sidereal  interval  before  subtracting 
from  M. 

The  true  mean  time  of  apparent  noon  is  found  by  applying 


ASTRONOMICAL  DETERMINATIONS  179 

to  12  o'clock  (the  apparent  time)  the  equation  of  time  for  the  given 
date  and  longitude.  The  equation  of  time  (with  directions  for 
applying)  is  found  in  the  Nautical  Almanac  for  Greenwich 
apparent  noon  of  the  given  date,  and  interpolated  for  the  given 
meridian.  The  true  mean  time  of  apparent  noon  is  then  reduced 
to  standard  time  (Art.  966),  or  sidereal  time  (Art.  99),  if  so  desired. 

By  comparing  the  observer's  time,  U,  with  the  corresponding 
true  time  of  apparent  noon,  the  error  of  the  observer's  timepiece 
at  apparent  noon  is  made  known. 

For  solar  observations  on  an  afternoon  and  following  morning, 
expressed  in  mean  time  units,  we  have  from  astronomy 

,,    ,   dd-t/tsui(f>    ,   tan  d 

"h 


in  which  L  is  the  observer's  time  at  apparent  midnight  (lower 
transit  of  sun),  d  and  dd  the  declination  and  hourly  change  for 
mean  midnight  of  initial  date,  and  the  other  quantities  remain 
as  before.  This  problem  is  worked  out  as  in  the  preceding  case 
except  that  d,  dd,  and  the  equation  of  time  must  be  interpolated 
for  twelve  hours  more  than  the  given  longitude,  and  the  clock 
error  is  determined  for  apparent  midnight  of  the  initial  date. 

For  star  observations  during  the  same  night,  taken  on  the  same 
star,  the  middle  time  represents  the  observer's  time  for  the  star's 
upper  transit.  The  true  sidereal  time  of  this  transit  equals  the 
star's  right  ascension,  as  given  in  the  Nautical  Almanac,  and  this 
is  changed  to  local  mean  time  (Art.  100),  and  thence  to  standard 
time  (Art  966),  if  so  desired. 

By  comparing  the  observer's  middle  time  with  the  true  time 
of  upper  transit,  the  error  of  the  observer's  timepiece  is  deter- 
mined for  the  moment  at  which  it  indicated  the  middle  time. 

For  star  observations  on  morning  and  evening  of  same  day,  taken 
on  the  same  star,  the  middle  time  represents  the  observer's  time 
for  the  star's  lower  transit.  The  true  sidereal  time  of  this  transit 
equals  the  star's  right  ascension  plus  twelve  hours,  and  .this  is 
changed  to  local  mean  time  (Art.  100),  and  thence  to  standard 
time  (Art.  966),  if  so  desired. 

By  comparing  the  observer's  middle  time  with  the  true  time 
of  lower  transit,  the  error  of  the  observer's  timepiece  is  determined 
for  the  moment  at  which  it  indicated  the  middle  time. 


180 


GEODETIC  SURVEYING 


EXAMPLE.— TIME  BY  EQUAL  ALTITUDES  OF  THE  SUN 


ALBANY,  N.  Y.,  May  10,  1911. 

Latitude    =42°  39'  12". 7  N. 

Longitude  =  73°  46'  42". 0  =  4*  55m 06s. 8  =  4.92  hrs.W.  of  Greenwich. 
Uses  75th  meridian  (Eastern  Standard)time  =  5.00  hrs.  W.  of  Greenwich. 
Local  time -Standard  time  =  5h  00m  00s.0-4h  55m  06s.8  =  4m  53s.  2 


Time.  d 

At  Greenwich  mean  noon  +17°  23'  56''. 7 
At  Greenwich  app.  noon 

Reduction  for  4. 92  hrs.  +       3    15    .8 

At  Albany  mean  noon  + 17°  27'  12" .  5 
At  Albany  app.  noon 


dd 

+  39".  87 

-  0    .15 

+  39".  72 


Eq.  of  Time 

3m418.2 
+    0  .6 

3m4ls.8 


Equation  of  time  subtractive  from  apparent  time(or\  given  date). 


0.73X4.92 
24 


— 0".15 


39.87  +  39.72 


=  39".  80 


0. 126X4. 92  =  0S. 62 
Sun.       App.  alt. 
f  45°  00' 


39.80X4.92   =195". 8 


Upper  limb  \  45 

[45 

r45 

Lower  limb  \  46 

I  46 


=  llh  56m178.5 
=   2h  52m34s.O 
=  2.88hrs. 
=43°  08'30".0 


20 
40 
40 
00 
20 


Watch,  A.M. 
8h  58m  22s 

00    18 

02 

05 
07 

09 


12 
14 
10 
05 


Watch,  P.M. 
2h  54m  13a 

52     18 

50 

47 

45 

43 


24 
20 
25 

29 


6)54h  22m  21s 


9h  03m43s.5 
(  +  12)  2     48    51  .5 


6)16h  53m098 
(  +  12)2h  48m51s.5 
9    03    43  .5 


2)23h  52m  358.0  2)5h  45m088.0 

2h  52m  348.0 


sn 


log. 

^>  (42°  39'  12".  7)  =  9.  9643882 
t  (43    08    30    .0)  =9.8349320 


173.5 


tan  d  (17°  27' 
tan  t  (43    08 


12' 
30 


.5) 
•  0) 


log. 

••  9. 4974948 
= 9. 9718084 


(1.3473) 


=  0.1294562 


(0.3355) 


=  9.5256864 


1.3473-0.3355  =  1.0118 


M 


56m  17s. 5 

-7.7 


dd  (39.72) 

t  (2.88) 

15     (a.c.) 

1.0118 


= 1 . 5990092 
=0.4593925 
=8.8239087 
=0.0050947 


56m098.8  (7s. 7) 

Local  apparent  noon  12h  00m  00s. 0 

Equation  of  time  3  41  .8 

Local  mean  time  at  apparent  noon  llh  56m  18s. 2 

Watch  time  at  apparent  noon  11  56  09  .8 

Watch  slow  by  mean  time  Om  08s .  4 

Reduction  to  standard  time  4  53  . 2 

Watch /as«  by  standard  time  4m448.8 


0.8874051 


ASTRONOMICAL  DETERMINATIONS  181 

103.  Time  by  SuiLand  Staf  Transits.  The  true  time  at  which 
any  heavenly  body  crosses  the  meridian  is  always  known;  in  the 
case  of  the  sun  the  upper  transit  is  apparent  noon,  the  mean 
time  of  which  is  determined  by  the  equation  of  time  (Art  96); 
in  the  case  of  a  star  the  sidereal  time  at  upper  transit  is  the 
same  as  the  star's  right  ascension;  and  (by  Arts.  966,  99,  and  100) 
sidereal  time,  mean  time  and  standard  time  are  mutually 
convertible.  If  the  observer  notes  his  own  clock  time  when  any 
heavenly  body  crosses  the  meridian,  the  error  of  his  timepiece 
is  made  apparent  by  comparison  with  the  corresponding  known 
true  time.  In  order  that  the  observation  may  be  made  it  is 
necessary  to  know  the  location  of  the  true  meridian  from  a  pre- 
vious azimuth  determination.  (Astronomers  have  other  ways 
of  obtaining  the  meridian.)  With  the  telescope  in  the  plane  of 
the  true  meridian,  and  set  at  a  suitable  vertical  angle,  it  is  only 
necessary  to  note  the  time  when  the  given  transit  occurs. 

103a.  Sun  Transits  with  Engineering  Instruments.  The  instru- 
ments used  for  determining  time  by  transits  of  the  sun  may  be 
the  ordinary  engineer's  transit  or  the  altazimuth  instruments  of 
Chapter  III.  A  prismatic  eyepiece  will  be  required  if  the 
meridian  altitude  exceeds  about  60°.  The  instrument  (and 
striding  level,  if  there  be  one)  should  be  in  good  adjustment. 
The  instant  at  which  the  advancing  edge  of  the  sun  reaches  the 
meridian  is  noted  with  the  telescope  direct,  and  the  instant  at 
which  the  following  edge  reaches  the  meridian  is  noted  with  the 
telescope  reversed,  the  mean  of  the  two  time  readings  being  the 
observer's  time  of  meridian  passage.  When  the  telescope  is 
reversed  it  will  be  necessary  to  revolve  the  instrument  on  its 
vertical  axis,  and  the  telescope  must  be  again  brought  into  the 
plane  of  the  meridian  by  sighting  at  the  meridian  mark  as  before. 
If  the  instrument  has  no  striding  level  the  plate  bubble  parallel 
to  the  horizontal  axis  of  the  telescope  is  to  be  kept  exactly  central 
while  each  observation  is  being  made.  If  the  instrument  has  a 
striding  level  it  must  not  be  reversed  when  the  telescope  is 
reversed,  but  the  bubble  must  be  kept  central,  as  before,  for 
each  observation.  If  the  instrument  has  three  leveling  screws 
it  should  be  set  with  two  screws  parallel  to  the  meridian  and 
the  bubble  kept  central  with  the  remaining  screw;  if  there  are 
four  leveling  screws,  place  one  pair  in  the  meridian  and  hold  the 
bubble  central  with  the  other  pair.  Time  determined  in  the 


182  GEODETIC  SURVEYING 

above  manner  should  not  be  in  error  by  more  than  a  second  with 
an  altazimuth  instrument,  nor  by  more  than  a  couple  of  seconds 
with  an  engineer's  transit. 

The  above  method  is  not  adapted  to  precise  time  determina- 
tions, so  that  when  the  larger  astronomical  instruments  are  avail- 
able the  observations  are  usually  made  on  the  stars. 

103b.  Star  Transits  with  Engineering  Instruments.  The 
instruments  used  by  the  engineer  for  determining  time  by  star 
transits  may  be  the  ordinary  transit  or  the  altazimuth  instru- 
ments of  Chapter  III.  The  instrument  (and  striding  level,  if 
there  be  one)  should  be  in  good  adjustment.  The  stars  have  no 
appreciable  diameter,  so  that  only  one  observation  is  obtained 
for  each  star.  Since  the  true  time  of  each  star  transit  will  be 
needed  in  the  reductions  it  is  desirable  to  tabulate  these  values 
beforehand,  in  order  to  be  ready  to  watch  for  each  transit  near 
the  proper  time,  as  a  star  occupies  only  about  a  minute  or  two 
in  crossing  the  field  of  view.  As  previously  explained  (Art.  97) 
the  sidereal  time  of  transit  for  each  star  is  the  same  as  its  right 
ascension;  if  the  observer's  timepiece  records  mean  or  standard 
time  it  will  be  necessary  to  reduce  the  sidereal  time  of  transit 
accordingly,  as  explained  in  Art.  100.  In  order  to  eliminate  instru- 
mental errors  the  stars  are  observed  in  pairs,  the  two  stars  of 
each  pair  having  about  the  same  declination;  the  second  star 
of  each  pair  is  then  observed  with  the  telescope  reversed.  The 
instructions  in  the  preceding  article  concerning  the  reversing  and 
releveling  of  the  instrument  must  be  strictly  adhered  to.  Only 
one  result  is  obtained  from  each  pair  of  stars,  the  average  true 
time  of  transit  for  each  pair  being  compared  with  the  middle 
observed  time  for  that  pair  to  obtain  the  clock  error  for  that 
instant  of  time.  Not  less  than  three  pairs  of  stars  should  be 
observed  and  the  results  averaged  If  the  clock  rate  is  not  known 
the  middle  times  for  the  several  pairs  should  not  differ  greatly, 
the  average  of  the  error  determinations  being  considered  as  the 
true  value  at  the  average  of  the  middle  times.  If  the  clock  rate 
is  known  the  several  error  determinations  are  first  reduced  to  the 
same  instant  of  time  before  averaging. 

Selection  of  stars.  If  several  pairs  of  stars  are  observed  it 
makes  no  difference  in  what  order  the  stars  come  to  the  meridian 
so  long  as  they  are  properly  paired  in  the  reductions.  If  the 
stars  are  so  selected  that  all  the  first  stars  of  the  several  pairs 


ASTRONOMICAL  DETERMINATIONS  183 

will  cross  the  meridian  beforg  any  of  the  second  stars,  but  one 
reversal  of  the  instrument  will  be  required;  this  will  be  the  case 
if  all  the  first  stars  have  less  right  ascensions  than  any  of  the 
second  stars.  In  order  to  have  ample  time  between  observations 
for  releveling,  etc.,  stars  should  not  be  selected  having  right 
ascensions  differing  by  less  than  about  five  minutes.  Having 
decided  on  the  period  of  the  night  during  which  it  is  desired  to 
make  the  observations,  the  mean  time  for  the  beginning  and  end 
of  this  period  must  be  converted  into  approximate  sidereal  time, 
and  stars  must  be  selected  whose  right  ascensions  lie  within  these 
limits.  The  approximate  sidereal  time  for  any  mean  time  instant 
is  found  by  adding  the  mean  time  interval  from  the  preceding 
noon  to  the  sidereal  time  of  Greenwich  mean  noon  for  the  same 
date.  Stars  near  either  pole  are  not  suitable  for  time  stars  on 
account  of  their  apparent  slow  movement  across  the  meridian; 
it  is  not  desirable  to  use  stars  whose  declination  is  more  than  60° 
either  way  from  the  equator.  On  account  of  the  uncertain  state 
of  the  atmosphere  at  low  altitudes  stars  should  not  be  selected 
which  will  cross  the  meridian  less  than  30°  above  the  horizon.  Thus 
in  40°  north  latitude  (see  Fig.  47,  page  166),  the  horizon  will  lie  50° 
south  of  the  equator,  and  hence  stars  should  not  be  taken  lying 
over  20°  south  of  the  equator,  so  that  for  this  latitude  the  stars 
selected  should  lie  between  60°  north  declination  and  20°  south 
declination.  The  altitude  of  any  star  while  crossing  the  meridian 
is  readily  obtained  when  it  is  remembered  that  the  meridian 
altitude  of  the  equator  equals  the  observer's  co-latitude,  and  that 
the  star's  distance  from  the  equator  is  given  by  its  declination. 
A  prismatic  eyepiece  will  be  required  for  meridian  altitudes 
exceeding  about  60°. 

It  is  best  to  use  the  brightest  stars  available  for  the  given 
time  and  place,  as  it  is  not  easy  to  identify  or  observe  the  fainter 
stars;  satisfactory  results  may  be  obtained  with  stars  ranging 
from  the  first  (brightest)  magnitude  to  about  the  fifth  magnitude, 
depending  on  the  size  of  the  instrument.  A  large  list  of  stars 
from  which  to  choose,  with  all  necessary  data,  will  be  found 
in  the  Nautical  Almanac. 

103c.  Star  Transits  with  Astronomical  Instruments.  The 
most  accurate  determinations  of  time  are  made  by  observing 
star  transits  with  large  portable  astronomical  transits  or  the 
still  larger  fixed  observatory  transits,  in  conjunction  with  an 


184  GEODETIC  SURVEYING 

astronomical  clock  beating  seconds  or  a  sidereal  chronometer 
beating  half  seconds.  A  portable  transit  is  illustrated  in  Fig.  48. 
A  chronograph  is  generally  used  in  the  observatory,  and 
sometimes  in  the  field,  for  recording  the  observations.  A  chrono- 
graph is  a  clock-like  device  for  moving  a  sheet  of  paper  uniformly 
under  a  pen  which  automatically  registers  each  second  as  indicated 
by  the  clock  or  chronometer;  by  breaking  an  electric  circuit 
the  observer  causes  the  pen  to  record  the  star  transits  on  the  same 
sheet  of  paper;  the  time  of  transit  is  then  obtained  very  accurately 
by  scaling  the  distance  from  the  nearest  recorded  second.  When 
the  chronograph  is  not  used  the  observer  listens  to  the  chronom- 
eter beats  and  estimates  the  time  of  each  transit  to  the 
nearest  tenth  of  a  second.  The  details  of  the  instruments 
used,  and  the  refinements  in  the  methods  of  observation 
and  computation,  are  beyond  the  scope  of  this  treatise,  but 
the  principles  involved  are  the  same  as  those  already  given. 
The  accuracy  attainable  is  to  about  the  nearest  one-hundredth 
part  of  a  second. 

104.  Choice  of  Methods.     Though  other  methods  have  been 
devised  for  determining  time,  those  above  given  are  the  ones  in 
most  general  use.    The  engineer  may  use  any  of  the  methods  from 
Art.  101  to  Art.  1036,  inclusive.     Engineers  generally  prefer  to 
work  in  the  daytime,  taking  their  observations  on  the  sun.     The 
transit  may  be  used,  but  the  sextant  is  to  be  preferred.  If  the 
transit  is  used  the  method  based  on  the  meridian  passage  of  the 
sun  (Art.  103a)  is  likely  to  be  the  most  satisfactory,  while  if  the 
sextant  is  used  the  method  of  equal  altitudes  (Arts.  102,  102a, 
1026)  will  generally  give  the  best  results.     Any  of  the  methods 
will  determine  the  true  time  as  closely  as  the  engineer  will  need 
it  in  any  of  his  operations. 

105.  Time  Determinations  at  Sea.    There  are  several  methods 
of  finding  local  time  at  sea,  the  method  by  single  altitudes  (Art. 
101)  being  most  commonly  used.     The  object  observed  may  be 
the  sun  or  one  of  the  brighter  stars.     The  observations  are  made 
with  the  sextant,  the  altitudes  being  measured  from  the  sea 
horizon.     This  horizon  is  not  the  true  horizon  on  account  of  the 
eye  of  the  observer  being  at  a  material  height  above  the  surface 
of  the  water.    The  result  of  this  condition  is  to  make  all  measured 
altitudes  too  large  by  an  angle  depending  on  the  height  of  the 
observer  and  known  as  the  dip  of  the  horizon.     The  correction 


ASTRONOMICAL  DETERMINATIONS 


185 


FIG.  48.— Portable  Transit. 
From  a  photograph  loaned  by  the  U.  S.  C.  and  G.  S. 


186  GEODETIC  SURVEYING 

for  dip  is  always  subtractive,  and  is  in  addition  to  the  corrections 
required  by  Art.  101&.  Its  value  is  given  by  the  formula 

log  D  =  1.7712700  +  i  log  h, 

in  which  D  is  the  dip  in  seconds  of  arc  and  h  is  the  observer's 
height  in  feet  above  the  sea.  The  latitude  required  in  the  for- 
mula of  Art.  1016  is  obtained  sufficient!}'  close  by  dead  reckoning 
from  the  nearest  observed  latitude.  Time  at  sea  may  be  deter- 
mined in  this  manner  with  a  probable  error  running  upwards 
from  a  few  seconds,  depending  on  the  circumstances  surrounding 
the  observations. 

LATITUDE 

106.  General  Principles.  The  latitude  of  a  point  on  the 
surface  of  the  earth  is  its  angular  distance  from  the  equator  in  a 
meridional  plane.  In  Fig.  49  the  ellipse  WNES  represents  a 
meridian  section  of  the  earth  (Arts.  65,  66,  67),  in  which  NS  is 
the  polar  axis,  or  minor  axis  of  the  ellipse;  WE,  the  equatorial 
diameter,  or  major  axis  of  the  ellipse;  n,  the  position  of  the 
observer;  nt  the  tangent  at  n;  ril,  the  normal  at  n,  it  being  noted 
that  the  normal  at  any  point  n  does  not  pass  through  the  center 
c  (except  when  n  is  at  the  poles  or  on  the  equator) ;  Znt  the  direc- 
tion of  the  plumb  line  at  n,  frequently  deviating  a  few  seconds 
(Art.  75)  from  the  direction  of  the  normal  nl]  Z,  the  zenith, 
or  intersection  of  the  direction  of  the  plumb  line  with  the  celestial 
sphere  (Art.  94). 

Astronomical  latitude  is  the  angular  distance  of  the  zenith 
from  the  equator,  or  the  angle  between  the  plumb  line  and  the 
equatorial  plane.  In  Fig.  49  the  astronomical  latitude  of  the 
point  n  would  be  shown  by  prolonging  the  line  Zn  to  an  intersec- 
tion with  the  line  WE,  the  intersection  commonly  falling  slightly 
to  one  side  of  the  point  I  and  making  the  angle  a  few  seconds  greater 
or  less  than  the  angle  </>.  The  latitude  as  determined  by  observa- 
tion is  always  the  astronomical  latitude.  Latitudes  obtained  at 
sea  are  of  this  kind. 

Geodetic  latitude  is  the  angle  between  the  normal  and  the 
equator;  in  Fig.  49  the  geodetic  latitude  of  the  point  n  is  the 
angle  </>.  The  geodetic  latitude  can  never  be  directly  observed, 
nor  can  the  deviation  of  the  plumb  line  be  found  by  direct  meas- 


ASTRONOMICAL  DETERMINATIONS 


187 


urement.  If,  however,  the  latitude  of  the  point  n  be  found 
by  computation  (Chapter  V)  from  the  astronomical  latitudes 
measured  at  various  other  triangulation  stations,  and  these 
values  be  averaged  in  with  its  own  astronomical  latitude,  the 
result  may  be  assumed  to  be  free  from  the  effects  of  plumb  line 


deviation  and  to  represent  the  true  geodetic  latitude.  In  geodetic 
work  geodetic  latitude  is  always  understood  unless  otherwise 
specified. 

Geocentric  latitude  is  the  angle  between  the  equator  and  the 
radius  vector  from  the  center  of  the  earth;  in  Fig.  49  the  geo- 
centric latitude  of  the  point  n  is  the  angle  /?.  The  geocentric 


188  GEODETIC  SURVEYING 

latitude  can  never  be  directly  observed.     It  is  computed  from  the 
geodetic  latitude  by  the  formula 


in  which  (Art.  69) 


62 

tan  8  =  -5  tan 
or 


log  ^~  =  9.9970504  -  10. 


At  the  equator  the  geodetic  and  geocentric  latitudes  are  each 
equal  to  zero.  At  the  poles  they  are  each  equal  to  90°.  At  any 
other  point  the  geocentric  latitude  is  less  than  the  geodetic 
latitude.  By  the  calculus  we  have, 

tan  (f)  (for  0  -  ft  =  max.)  «  p     or     0  =  45°  05'  50".21; 

tan  ft  (for  $  -  ft  =  max.)  =  -,     or    ft  =  *A    54    09  .79; 

or  a  maximum  difference  of  11'  40".42.  The  popular  conception 
of  latitude  is  geocentric  latitude,  but  published  latitudes  are 
usually  astronomical  latitudes  or  geodetic  latitudes. 

107.  Latitude  from  Observations  on  the  Sun  at  Apparent 
Noon.  Latitude  sufficiently  close  for  many  purposes  may  be 
obtained  by  measuring  the  altitude  of  the  sun  at  apparent  noon, 
or  the  moment  when  it  crosses  the  meridian.  The  local  mean 
time  of  apparent  noon  is  found  by  applying  to  12  o'clock  (the 
apparent  time)  the  equation  of  time  as  taken  from  the  Nautical 
Almanac  for  the  given  date,  interpolating  for  the  given  meridian; 
the  corresponding  standard  time  may  then  be  found  by  Art.  96a. 
If  the  correct  time  is  not  known  the  altitude  is  measured 
when  it  attains  its  greatest  value,  which  soon  becomes  evident 
to  the  observer  who  is  following  it  up.  A  good  observer  can  obtain 
an  observation  on  each  limb  of  the  sun  before  there  is  any  appre- 
ciable change  of  altitude,  the  mean  of  the  readings  being  the 
observed  altitude  for  the  center;  if  only  one  limb  is  observed 
the  reading  must  be  reduced  to  the  center  by  applying  a  correc- 
tion for  semi-diameter  as  found  in  the  Nautical  Almanac  for  the 
given  date,  the  result  being  the  observed  altitude.  In  either 
case  ithe  observed  altitude  is  too  large  on  account  of  refraction, 
and  must  be  corrected  by  an  .amount  which  may  be  taken  from 


ASTRONOMICAL  DETERMINATIONS  189 

Table  VIII  for  the  given  observed  altitude.  Theoretically  all 
solar  altitudes  are  measured  too  small  on  account  of  parallax 
(due  to  the  observer  not  being  at  the  center  of  the  earth),  the 
necessary  correction  being  equal  to  8".9  multiplied  by  the  cosine 
of  the  observed  altitude.  The  correction  for  parallax  is  a  useless 
refinement  with  the  engineer's  transit,  but  may  be  applied,  if 
desired,  when  a  sextant  or  altazimuth  instrument  is  used. 

The  observation.  Single  altitudes  of  the  sun  may  be  measured 
with  a  transit  or  with  an  altazimuth  instrument,  but  a  pris- 
matic eyepiece  will  be  required  if  the  altitude  exceeds  about  60°. 
The  instrument  must  be  very  carefully  leveled  at  the  moment  of 
taking  the  observation,  and  if  two  readings  can  be  secured  the 
second  reading  should  be  taken  on  the  other  limb  of  the  sun  with 
the  telescope  reversed  and  the  instrument  carefully  releveled, 
so  as  to  eliminate  the  instrumental  errors.  If  only  one  reading 
is  secured  it  should  be  corrected  for  index  error  if  one  exists.  If 
the  altitude  is  not  greater  than  about  60°  an  artificial  horizon 
may  be  used  and  the  double  altitude  measured  with  either  of  the 
above  instruments  or  a  sextant.  If  a  transit  or  altazimuth 
instrument  is  used  it  is  not  reversed  on  any  of  the  observa- 
tions, and  it  must  not  be  releveled  between  the  pointing  to 
the  sun  and  the  pointing  to  its  reflected  image.  If  a  sextant  is 
used  the  correction  for  index  error  must  be  applied. 

The  computation.  Having  applied  the  appropriate  correc- 
tions to  the  measured  altitude,  as  described  above,  the  true 
altitude  of  the  sun  is  obtained  within  the  capacity  of  the  instru- 
ment used.  This  value  being  subtracted  from  90°  gives  the  zenith 
distance  of  the  sun.  The  declination  of  the  sun  is  taken  from  the 
Nautical  Almanac  for  the  given  date  and  meridian,  and  this 
value  is  the  distance  of  the  sun  from  the  equator.  Knowing  thus 
the  distance  from  the  equator  to  the  sun,  and  from  the  sun  to 
the  zenith,  an  addition  or  subtraction  (as  the  case  requires) 
gives  the  zenith  distance  of  the  equator,  and  this  value  (Art.  106) 
is  the  observer's  latitude.  If  an  ordinary  transit  is  used  the 
latitude  thus  obtained  should  be  correct  to  the  nearest  minute. 
If  a  sextant  or  an  altazimuth  instrument  is  used  the  result  is 
generally  much  closer  to  the  truth.  Theoretically  the  result 
should  be  as  accurate  as  the  instrument  will  read,  but  there  is 
always  a  doubt  as  to  the  precise  value  of  the  refraction,  and  the 
latitude  obtained  is  subject  to  the  same  uncertainty. 


190  GEODETIC  SURVEYING 

108.  Latitude  by  Culmination  of  Circumpolar  Stars.  Stars 
having  a  polar  distance  (90°  —declination)  less  than  the  observer's 
latitude  never  set,  but  appear  to  revolve  continuously  around 
the  pole,  and  are  hence  called  circumpolar  stars.  Such  stars 
cross  the  observer's  meridian  twice  every  day,  once  above  the 
pole  (upper  culmination)  and  once  below  the  pole  (lower  culmina- 
tion). By  referring  to  Fig.  47,  page  166,  it  will  be  seen  that  the 
latitude  of  any  place  is  always  the  same  as  the  altitude  of  the 
elevated  pole.  By  observing  the  altitude  of  a  close  circumpolar 
star  at  either  upper  or  lower  culmination,  and  combining  the 
result  (minus  correction  for  refraction,  Table  VIII)  with  the  star's 
polar  distance  (added  for  lower  culmination,  subtracted  for 
upper  culmination),  the  altitude  of  the  elevated  pole  is  obtained, 
and  hence  the  observer's  latitude.  The  polar  distance  must  be 
based  on  the  declination  for  the  given  date  as  found  in  the 
Nautical  Almanac.  The  latitude  as  thus  determined  is  much 
more  reliable  than  that  obtained  by  solar  observations. 

In  the  northern  hemisphere  the  best  star  to  observe  is  Polaris 
(«  Ursae  Minoris),  on  account  of  its  brightness  (2nd  magnitude) 
and  its  small  polar  distance  (about  1°  10'  in  1911).  About  the 
middle  of  the  year  both  culminations  of  Polaris  occur  during 
daylight  hours,  rendering  it  unsuitable  for  observation.  The  next 
best  star  to  observe  is  51  Cephei,  which  also  has  a  small  polar  dis- 
tance (about  2°  48'  in  1911),  but  whose  brightness  (5th  magnitude) 
is  not  equal  to  that  of  Polaris.  As  these  two  stars  differ  about 
five  and  one-half  hours  in  right  ascension,  at  least  one  of  them 
must  culminate  during  the  hours  of  darkness.  The  sidereal  time 
of  upper  culmination  for  either  star  is  the  same  as  its  right  ascen- 
sion (the  exact  value  for  the  given  date  being  taken  from  the 
Nautical  Almanac),  and  this  is  converted  into  mean  time  by 
Art.  100.  By  a  study  of  Fig.  50,  which  shows  the  arrangement  of 
a  number  of  stars  in  the  vicinity  of  the  north  pole  of  the  heavens, 
it  will  not  be  difficult  to  identify  Polaris  and  51  Cephei.  The 
polar  distances  of  these  stars  are  so  small  that  but  little  change 
of  altitude  occurs  when  they  are  near  the  meridian,  so  that  several 
observations  may  be  obtained  and  averaged.  If  the  observations 
are  taken  within  five  minutes  each  side  of  the  meridian  the  error 
in  assuming  the  altitudes  unchanging  will  not  exceed  I"  with 
Polaris  and  2".5  with  51  Cephei,  and  may  be  ignored  when  observ- 
ing with  engineering  instruments.  Within  fifteen  minutes  either 


ASTRONOMICAL  DETERMINATIONS  191 


.° 


\» 

<  \\ •    V 

\          *\  \  o  • 

*    \  A 

•a 
'—  .^  M  I 

^  s 

•   <^ 

/x*  Q 

•       *    •        -1  5 


/'  <s  a  -a  * 

/       ••  fe  ^    «*  o 

s  / 

/    • 


_,i  ,a^°.f         s . 


•  t  -/      fc  .2  5 

/  /  ^*        HI  *?      *  o 

/  /         •  ^ 

o>  • 

\  s  •-  .    s       '.  •  •     • 

•^   O  -.~ 


192  GEODETIC  SURVEYING 

way  from  meridian  passage  the  change  in  altitude  (within  1" 
error)  may  be  found,  if  desired,  by  multiplying  the  square  of  the 
time  (in  minutes)  from  culmination  by  0".044  for  Polaris  and 
0".104  for  51  Cephei.  If  this  correction  is  applied  it  is  to  be  added 
to  observations  near  upper  culmination  and  subtracted  from 
observations  near  lower  culmination,  to  obtain  the  corresponding 
culminating  altitude. 

In  making  the  observation  the  altitude  maybe  directly  measured 
with  a  transit  or  an  altazimuth  instrument.  In  order  to  eliminate 
instrumental  errors  at  least  two  readings  should  be  averaged 
together,  one  taken  with  telescope  direct  and  one  with  telescope 
reversed.  The  instrument  must  be  releveled  after  reversing, 
as  it  is  necessary  to  have  the  bubbles  exactly  central  at  the  moment 
each  reading  is  taken.  If  by  any  accident  only  one  reading  is 
secured  it  must  be  corrected  for  index  error,  if  one  exists.  The 
two  readings  should  be  obtained  as  near  together  and  as  near 
culmination  as  the  skill  of  the  observer  will  permit;  two  readings 
not  over  three  minutes  each  way  from  the  meridian  are  easily 
'obtained.  A  better  result  will  be  obtained  if  four  readings  are 
averaged  together,  taking  one  direct  reading,  then  two  reversed 
readings,  and  then  one  direct  reading,  both  bubbles  being  kept 
exactly  central  while  taking  each  reading;  this  program  is 
easily  accomplished  within  five  minutes  each  side  of  the  meridian. 
If  an  artificial  horizon  is  available  it  is  better  to  measure  the  double 
altitude  between  the  star  and  its  image  in  the  mercury,  using 
either  of  the  above  instruments  or  a  sextant.  Angles  measured 
with  a  sextant  are  always  corrected  for  index  error  and  sometimes 
for  eccentricity.  If  a  transit  or  altazimuth  instrument  is  used  the 
double  altitude  is  obtained  by  reading  on  the  star  and  then  on 
its  image,  without  reversing  or  releveling  between  the  pointings. 
Two  such  double  altitudes  are  easily  obtained  within  three  minutes 
each  way  from  the  meridian,  using  either  of  these  instruments 
or  a  sextant.  Latitudes  obtained  by  the  methods  of  this  article 
should  theoretically  be  correct  within  the  reading  capacity  of 
the  instrument,  but  may  be  further  in  error  on  account  of  the 
uncertainties  of  refraction. 

109.  Latitude  by  Prime  Vertical  Transits.  Stars  whose 
declination  is  less  than  the  observer's  latitude  apparently  cross 
the  prime  vertical  (true  east  and  west  vertical  plane)  twice  dur- 
ing each  revolution  of  the  earth  on  its  axis.  If  the  time  elapsing 


ASTRONOMICAL  DETERMINATIONS 


193 


between  the  east  and  west  transit  of  any  star  is  noted  the  observ- 
er's latitude  may  be  found  by  com- 
putation. Referring  to  Fig.  51,  P  is 
the  elevated  pole  of  the  celestial 
sphere;  PZS',  the  observer's  meridian; 
Z,  the  observer's  zenith;  SZS",  the 
prime  vertical;  SS'S",  the  star's  ap- 
parent path;  PS,  the  star's  polar 
distance;  and  PZ}  the  observer's  co- 
latitude.  In  the  spherical  triangle 
PZS,  right-angled  at  Z,  the  side  PS 
and  the  angle  SPZ  are  known;  the 
side  PS  being  the  star's  polar  distance, 
and  the  angle  SPZ  equal  to  half  the 

elapsed  time  changed  to  angular  units  by  multiplying  by  15. 
Hence,  solving  for  the  latitude  </>,  we  have 

.       tan  § 

tan  0  = . 

cos  t 

In  this  method  the  uncertainties  of  refraction  are  largely  elim- 
inated because  the  times  of  transit  are  observed  instead  of  the 
altitudes.  The  success  of  the  method  depends  on  the  precision 
with  which  the  meridian  is  determined  and  the  prime  vertical 
located  therefrom,  and  the  accuracy  with  which  the  telescope 

is  made  to  describe  a  vertical  plane. 
The  method,  though  not  much  used 
in  the  United  States,  is  one  of  the 
best,  and  with  suitable  instruments 
and  refinements  will  determine  lati- 
tude within  a  fraction  of  a  second.  If 
a  close  determination  of  latitude  has 
to  be  made  with  an  altazimuth  instru- 
ment without  a  micrometer  eyepiece, 
but  which  is  furnished  with  a  good 
stridingleveljthismethod  will  probably 
give  better  results  than  any  other. 

110.  Latitude    with    the    Zenith 
Telescope.     This  method  (otherwise 
known     as     the    Harrebow-Talcott 
method)  is  the  one  which  the  U.  S.  Coast  and  Geodetic  Survey 


194  GEODETIC  SURVEYING 

always  uses  for  the  precise  determination  of  latitude,  the  probable 
error  of  the  results  being  readily  kept  below  a  tenth  of  a  second. 
Referring  to  Fig.  52,  page  193,  PEP'E'  is  a  meridian  section  of  the 
celestial  sphere;  PP',  the  polar  axis;  EE',  the  equator;  C,  the 
observer;  Z,  the  zenith;  S  and  S',  two  stars  with  nearly  equal 
(within  about  15')  but  opposite  meridian  zenith  distances,  and 
with  a  sufficient  difference  of  right  ascension  to  enable  each  one 
to  be  observed  in  turn  as  it  crosses  the  meridian. 

Let  0  =  EZ  =  observer's  latitude; 

d  =  ES  =  declination  of  S  (from  Nautical  Almanac) ; 
d'=  ES' =  declination  of  S'  (from  Nautical  Almanac); 
z  =  apparent  zenith  distance  of  S; 
z'  —  apparent  zenith  distance  of  /S'; 
r  =   refraction  correction  for  z  (from  Table  VIII) ; 
r'  =  refraction  correction  for  z'  (from  Table  VIII) ; 
then 

z  +  r    =  ZS  =  true  zenith  distance  of  S; 
z'  +  r'  =  ZS'  =  true  zenith  distance  of  £'; 
whence 

</>  =  d  +  z  +  r 
=  *'-  (z'+r') 


2$  =  (d  +  3')  +  (z-zf)  +  (r-r')' 
and  we  have  for  the  latitude 

-  m  +  <?')  +  (*  -  *' 


In  this  equation  the  quantities  (<?+  d')  and  (r—  rf)  are  known, 
so  that  it  is  only  necessary  to  obtain  (z—  z')  by  observation  to 
determine  the  latitude.  The  quantity  (z  —  z')  is  the  difference 
between  the  zenith  distances  of  the  two  stars  S  and  S',  and  if 
this  quantity  is  not  over  about  15'  it  can  be  measured  with  great 
accuracy  by  means  of  the  zenith  telescope  (see  Fig.  53).  The 
instrument  illustrated  has  an  aperture  of  about  three  inches, 
a  focal  length  of  nearly  four  feet,  and  a  magnifying  power  of  100. 
The  telescope  being  set  at  a  proper  vertical  angle  for  a  given  pair 
of  stars  is  not  changed  thereafter,  but  each  star  is  brought  into 
the  field  of  view  by  revolving  the  instrument  on  its  vertical 
axis,  and  the  difference  of  zenith  distance  is  measured  entirely 


FIG.  53.— Zenith  Telescope. 

From  a  photograph  loaned  by  the  U.  S.  C.  and  G.  S. 


196  GEODETIC  SURVEYING 

with  the  micrometer  eyepiece.  Many  pairs  of  stars  are  observed, 
and  many  refinements  in  observation  and  computation  are  required 
in  the  highest  grade  of  work.  For  a  complete  discussion  of  the 
method  the  reader  is  referred  to  Appendix  No.  7,  Report  for 
1897-98,  U.  S.  Coast  and  Geodetic  Survey.  An  altazimuth 
instrument  with  a  micrometer  eyepiece  will  give  very  good 
results  by  the  above  method,  if  used  with  proper  precautions. 

111.  Latitude  Determinations  at  Sea.     Many  methods  have 
been  devised  for  determining  latitude  at  sea.     Greenwich  time 
may  or  may  not  be  required,  according  to  the  method  used, 
but  is  generally  available  from  the  ship's  chronometers.     In  any 
case  the  observation  consists  in  measuring  with  the  sextant  the 
altitude  of  one  or  more  of  the  heavenly  bodies  above  the  sea 
horizon.     All  such  altitudes  are  reduced  to  the  true  horizon  by 
applying  a  correction  for  dip,  as  explained  in  Art.  105,  this  cor- 
rection being  in  addition  to  any  others  which  the  observation 
requires   to   determine   the   true   altitude.     The   most   common 
observation  for  latitude  is  for  the  altitude  of  the  sun  at  apparent 
noon,  as  explained  in  Art.  107.     The  meridian  altitude  of  the  pole 
star  or  other  bright  star  is  also  often  observed,  the  result  in  either 
case   being  worked  out  as  explained  for  circumpolar  stars  in 
Art.  108.     The  error  of  a  latitude  determination  at  sea  may  range 
upwards  from  a  fraction  of  a  mile,  depending  on  the  circumstances 
surrounding  the  observation. 

112.  Periodic  Changes  in  Latitude.     It  is  now  known  that  tlie 
earth  has  a  slight  wabbling  motion  with  respect  to  the  axis  about 
which  it  rotates.      In  consequence  of  this  motion  the  north  and 
south  poles  do  not  occupy  a  fixed  position  on  the  surface  of  the 
earth,  but  each  one    apparently  revolves   about  a  fixed  mean 
point  in  a  period  of  about  425  days.     The  distance  between 
the  actual  pole  and  the  mean  point  is  not  constant,  but  varies 
(during  a  series  of  revolutions)  between  about  0".16  (16.3  ft.), 
and  about  0".36   (36.6  ft.).     As  the  equator  necessarily  shifts 
its  position  in  accordance  with  the  movement  of  the  poles,  it 
follows  that  the  latitude  at  every  point  on  the  surface  of  the  earth 
is  subject  to  a  continual  oscillation  about  its  mean   value,   the 
successive  oscillations  being  of  different  extent  and  ranging  from 
0".16  to  0".32  each  way  from  the  middle.     In  precise  latitude 
work,  therefore,  the  date  of  the  determination  is  an  essential 
part  of  the  record. 


ASTRONOMICAL  DETERMINATIONS  197 

L6NGTTUDE 

113.  General  Principles.  The  longitude  of  any  point  on  the 
surface  of  the  earth  is  the  angular  distance  of  the  meridian  of  that 
point  from  a  given  reference  meridian,  being  positive  when  reckoned 
westward  and  negative  when  reckoned  eastward.  The  meridian 
of  Greenwich  has  been  universally  adopted  (since  1884)  as  the 
standard  reference  meridian  of  the  world,  but  other  meridians 
(Washington,  Paris,  etc.)  are  often  used  for  special  work.  Since 
time  is  measured  by  the  uniform  angular  movement  of  the  earth 
on  its  axis  (west  to  east),  it  follows  that  longitude  may  be 
expressed  equally  well  in  either  angular  units  or  time  units.  As 
360°  of  arc  correspond  to  twenty-four  hours  of  time  (mean  or 
sidereal,  Art.  95),  the  change  from  the  angular  to  the  time  system 
is  evidently  made  by  dividing  by  15,  and  vice  versa;  thus  the 
longitude  of  Washington  west  from  Greenwich  may  be  written 
as  77°  03'  56".7,  or  5h  08m  15S.78,  as  preferred.  At  the  same 
absolute  instant  of  time  the  true  local  time  of  any  station  differs 
from  the  true  local  time  of  any  other  station  by  the  angular 
divergence  (expressed  in  time  units)  of  the  meridians  of  these 
two  stations;  the  difference  of  longitude  of  any  two  stations, 
therefore,  is  identical  with  the  difference  of  local  time.  At  the 
same  instant  of  time,  the  difference  between  the  local  mean  time 
and  the  sidereal  time  at  any  station  is  the  same  for  all  points  in 
the  world,  so  that  the  difference  of  local  time  between  any  two 
given  stations  is  always  numerically  the  same  whether  the  com- 
parison is  based  on  local  mean  time  or  sidereal  time.  From  the 
nature  of  the  case,  it  is  evident  that  standard  time  (Art.  96a) 
bears  no  relation  to  the  longitude  of  a  station. 

Longitude  as  described  above  is  geodetic  longitude.  Longitude 
obtained  from  observations  on  heavenly  bodies,  or  astronomical 
longitude,  is  identical  with  geodetic  longitude  except  where  local 
deviation  of  the  plumb  line  (Art.  75)  exists.  The  geodetic  long- 
itude of  a  point  can  never  be  directly  observed,  nor  can  the  devia- 
tion of  the  plumb  line  be  found  by  direct  measurement.  If, 
however,  the  longitude  of  any  point  be  found  by  computation 
(Chapter  V)  from  the  astronomical  longitudes  measured  at 
various  other  triangulation  stations,  and  these  values  be  averaged 
in  with  its  own  astronomical  longitude,  the  result  may  be  assumed 
to  be  free  from  the  effects  of  plumb  line  deviation  and  to  represent 


198  GEODETIC  SURVEYING 

the  true  geodetic  longitude.     In  geodetic  work  geodetic  longitude 
is  always  understood  unless  otherwise  specified. 

The  longitude  of  any  given  point  is  ordinarily  obtained  by 
finding  how  much  it  differs  from  that  of  some  other  point  whose 
longitude  has  already  been  well  determined.  The  finding  of 
this  difference  of  longitude  is  essentially  the  finding  of  the  dif- 
ference of  local  time  between  the  two  points,  the  westerly 
point  having  the  earliest  time,  and  vice  versa.  The  local  time 
is  found  by  the  methods  heretofore  given,  and  the  comparison 
is  made  as  about  to  be  explained. 

114.  Difference   of  Longitude  by  Special  Methods.     These 
methods  are  rarely  used  any  more,  but  are  of  considerable  scientific 
interest,  and  hence  are  here  briefly  mentioned. 

By  special  phenomena.  Certain  astronomical  phenomena, 
such  as  the  eclipses  of  Jupiter's  satellites,  occur  at  the  same  instant 
of  time  as  seen  at  any  point  on  the  earth  from  which  they  may 
be  visible.  These  eclipses  usually  occur  several  times  in  the  course 
of  a  month,  the  Washington  mean  time  of  the  event  being  given 
in  the  Nautical  Almanac.  The  observer  notes  the  true  local  time 
at  which  the  eclipse  occurs,  the  error  and  rate  of  his  timepiece 
having  been  previously  determined.  The  difference  between 
the  Washington  mean  time  and  the  local  mean  time  of  the  eclipse 
is  the  observer's  longitude  from  Washington.  Eclipses  of  the 
moon  may  also  be  used  in  the  same  manner.  Longitude  obtained 
by  these  methods  is  apt  to  be  several  seconds  of  time  in  error, 
or  a  minute  or  more  in  arc. 

By  flash  signals.  Two  observers,  having  obtained  their  own 
local  time  by  proper  observations,  may  each  note  the  reading  of 
their  own  clock  at  the  same  instant  of  time,  this  instant  being 
determined  by  an  agreed  signal  visible  to  both.  Such  a  signal 
may  be  the  flash  of  a  heliotrope  by  day,  or  any  suitable  light 
signal  by  night.  The  difference  of  local  time  is  then  the  difference 
of  longitude.  The  error  by  this  method  may  be  kept  below  a 
second  of  time  by  averaging  the  results  of  a  number  of  signals. 
This  method  usually  requires  one  or  more  intermediate  stations 
to  be  established  to  overcome  the  lack  of  intervisibility,  and  is 
generally  an  expensive  one. 

115.  Longitude  by  Lunar  Observations.     If  an  observer  notes 
his  true  local  time  (expressed  as  mean  time)  for  any  particular 
position  of  the  moon,  and  obtains  from  the  Nautical  Almanac 


ASTRONOMICAL  DETERMINATIONS  199 

the  Greenwich  mean  time  whfjn  the  moon  occupied  such  a  posi- 
tion, the  longitude  from  Greenwich  is  given  by'the  corresponding 
difference  of  time.  Many  methods  have  been  devised  on  this 
basis,  requiring  laborious  computations  in  their  application,  and 
in  many  of  the  methods  not  leading  to  very  accurate  results. 
Lunar  methods  are  therefore  not  generally  used  except  on  long 
sea  voyages  or  long  exploration  trips.  A  few  of  the  methods  are 
given  below,  but  only  in  the  roughest  outline. 

By  lunar  distances.  The  angle  between  a  star,  the  center 
of  a  planet,  or  the  near  edge  of  the  sun,  and  the  illuminated  edge 
of  the  moon  may  be  measured  by  a  sextant,  and  reduced  to 
what  it  would  have  been  if  it  had  been  observed  at  the  center  of 
the  earth  and  measured  to  the  center  of  the  moon.  The  Green- 
wich time  of  this  position  can  be  determined  from  the  Nautical 
Almanac  and  compared  with  the  local  time  at  which  the  observa- 
tion was  made.  The  accuracy  attainable  is  about  five  seconds 
of  time. 

By  lunar  culminations.  The  local  time  of  meridian  passage 
of  the  moon's  illuminated  limb  may  be  noted,  expressed  as  sidereal 
time  and  corrected  for  semi-diameter,  giving  the  moon's  right 
ascension  at  the  given  instant,  and  Greenwich  mean  time  for 
this  value  of  the  right  ascension  be  compared  with  the  observed 
local  time.  The  accuracy  attainable  is  about  five  seconds  of  time. 

By  lunar  occultations.  The  occupation  (covering)  of  a  star 
by  the  moon  may  be  observed,  noting  the  local  time  of  immersion 
(disappearance),  or  emersion  (reappearance),  or  both,  in  which 
case  the  apparent  right  ascension  of  the  corresponding  edge  of 
the  moon  at  the  given  instant  is  the  same  as  the  right  ascension 
of  the  given  star.  When  proper  correction  has  been  made  for 
refraction,  parallax,  semi-diameter,  etc.,  the  true  right  ascension 
becomes  known  for  the  given  instant,  and  the  corresponding 
Greenwich  time  is  compared  as  before  with  the  local  observed  time. 
This  method,  with  the  exception  of  telegraphic  methods,  is  one  of 
the  best  that  is  known  for  longitude  work.  When  a  number 
of  such  determinations  are  averaged  together,  an  accuracy  within 
a  second  of  time  is  attainable. 

116.  Difference  of  Longitude  by  the  Transportation  of  Chro- 
nometers. When  this  method  is  used  a  number  of  chronometers 
(from  5  to  50)  are  carried  back  and  forth  (from  about  5  round  trips 
upwards)  between  the  two  points  whose  difference  of  longitude 


200  GEODETIC  SURVEYING 

is  desired.  On  reaching  each  station  the  traveling  chronometers 
are  compared  with  the  local  chronometers.  The  errors  of  the 
local  chronometers  are  determined  astronomically  at  or  near 
the  time  of  comparison.  The  various  values  thus  obtained  for  the 
difference  of  time  between  the  two  stations  are  averaged  together 
and  the  result  taken  as  the  difference  of  longitude.  Owing  to 
the  fact  that  each  round  trip  furnishes  two  determinations  that 
are  oppositely  affected  by  similar  errors,  and  also  to  the  refinements 
of  method  and  reduction  that  are  used  in  practice,  the  errors 
due  to  chronometer  rates  and  irregularities  are  largely  eliminated 
from  the  average  result.  The  accuracy  attainable  (in  time  units) 
may  range  between  a  few  tenths  of  a  second  and  less  than  a  single 
tenth  of  a  second,  depending  on  the  distance  between  stations, 
the  number  of  trips  made,  and  the  number  of  chronometers 
transported.  Longitude  determinations  by  this  method  are  now 
rarely  made,  except  where  telegraphic  connection  is  not  available. 
In  order  to  make  an  accurate  comparison  of  two  mean  time 
chronometers  each  one  is  independently  compared  with  the  same 
sidereal  chronometer,  and  two  sidereal  chronometers  are  sim- 
ilarly compared  by  mutual  reference  to  a  mean  time  chronometer. 
Sidereal  chronometers  continually  gain  on  mean  time  chronom- 
eters, the  beats  or  ticks  (half  seconds)  gradually  receding  from  and 
approaching  a  coincidence  that  occurs  about  every  three  minutes. 
When  the  beats  exactly  coincide  the  chronometers  differ  precisely 
by  the  value  in  half  seconds  indicated  by  the  subtraction  of  their 
face  readings.  As  the  ear  can  be  trained  to  detect  a  lack  of  coin- 
cidence as  small  as  the  one-hundredth  part  of  a  second,  a  com- 
parison can  be  made  with  this  degree  of  precision. 

117.  Difference  of  Longitude  by  Telegraph.  Where  tele- 
graphic connection  can  be  established  between  two  stations  it 
furnishes  the  best  means  of  exchanging  time  signals,  both  on 
account  of  the  great  accuracy  attainable  and  the  comparative 
inexpensiveness.  Difference  of  longitude  obtained  in  this  manner 
can  be  made  more  accurate  than  is  possible  by  any  other  known 
method.  The  lines  of  the  telegraph  companies  ramify  in  all 
directions,  and  the  temporary  use  of  a  suitable  wire  can  usually 
be  obtained  at  reasonable  cost,  so  that  it  is  only  necessary  to 
erect  short  connecting  lines  between  the  observing  stations  and  the 
telegraph  stations.  The  most  important  applications  of  the 
method  are  as  outlined  below. 


ASTRONOMICAL  DETERMINATIONS  201 

By  standard  time  signal^  This  method  furnishes  a  quick 
means  for  an  approximate  longitude  determination.  Standard 
time  can  be  obtained  at  any  telegraph  station  with  a  probable 
error  of  less  than  a  second.  The  observer's  true  local  mean  time 
is  obtained  by  any  of  the  simpler  methods  of  observation.  The 
difference  of  these  times  is  the  difference  of  longitude  between 
the  given  standard  time  meridian  and  the  meridian  of  the  ob- 
server's station. 

By  star  signals.  The  difference  of  longitude  of  any  two 
stations  is  identical  with  the  sidereal  time  which  elapses  between 
the  transit  of  any  given  star  over  the  meridian  of  the  easterly 
station,  and  the  transit  of  the  same  star  over  the  meridian  of  the 
westerly  station;  so  that  it  is  only  necessary  to  observe  how  long 
it  takes  for  any  star  to  pass  between  the  meridians  of  two  stations 
to  know  their  difference  of  longitude.  In  making  use  of  this 
principle  a  chronograph  (Art.  103c)  is  placed  at  each  station, 
and  these  chronographs  are  connected  by  a  telegraph  line.  A 
break-circuit  chronometer,  which  may  be  placed  anywhere  in 
this  line,  records  its  beats  on  both  chronographs.  As  the  selected 
star  crosses  the  meridian  of  the  easterly  observer  he  records  this 
instant  of  time  on  both  chronographs  by  tapping  his  break- 
circuit  signal  key.  When  the  same  star  crosses  the  meridian  of  the 
westerly  observer  he  likewise  records  this  new  instant  of  time 
on  both  chronographs.  Each  chronograph,  therefore,  contains 
a  record  of  the  time  between  transits,  but  the  records  are  not 
identical,  as  it  takes  time  for  the  signals  to  pass  between  the 
stations;  in  other  words,  each  signal  is  recorded  a  little  later  on 
the  distant  chronograph  than  it  is  on  the  home  chronograph. 
The  record  of  the  easterly  chronograph  thus  becomes  too  great, 
and  the  record  of  the  westerly  one  correspondingly  too  small; 
but  the  mean  of  the  two  records  eliminates  this  error  and  gives 
(when  corrected  for  chronometer  rate)  the  true  difference  of 
longitude  between  the  stations.  In  actual  work  the  transits  of 
many  stars  are  observed  at  each  station,  so  as  to  obtain  an  average 
value  for  the  difference  of  longitude.  The  accuracy  attainable 
is  about  0.01  of  a  second  of  time.  This  method  is  one  of  the 
best,  and  was  formerly  largely  used  by  the  Coast  Survey.  The 
objection  to  the  method  is  the  difficulty  of  securing  the  monopoly 
of  the  telegraph  line  during  the  long  period  while  the  observa- 
tions are  in  progress,  so  that  it  is  no  longer  much  in  use. 


202  GEODETIC  SURVEYING 

By  arbitrary  signals.  This  is  the  standard  method  of  the 
Coast  Survey  at  the  present  time,  and  requires  the  use  of  the 
telegraph  line  for  only  a  few  minutes  during  an  arbitrary  period 
(previously  agreed  upon)  on  each  night  that  observations  are  in 
progress.  In  this  method  a  chronometer  and  chronograph  are 
installed  at  each  station,  and  each  chronometer  records  its  beats 
on  the  home  chronograph  only.  Each  observer  makes  his  own 
time  observations,  which  are  likewise  recorded  on  his  own  chrono- 
graph alone.  Observations  at  each  station  are  taken  both  before 
and  after  the  exchange  of  signals  in  order  to  determine  the  cor- 
responding chronometer's  rate  as  well  as  its  error.  As  far  as 
possible  the  same  stars  are  observed  at  each  station,  in  order 
to  avoid  introducing  errors  of  right  ascension.  In  the  most 
precise  work  the  observers  exchange  places  on  successive  nights, 
in  order  to  eliminate  the  effects  of  personal  equation,  and  numerous 
other  refinements  are  introduced.  The  chronograph  sheet  at 
each  station  enables  the  true  time  at  that  station  to  be  computed 
for  any  instant  within  the  range  of  the  record,  and  the  difference 
of  these  true  times  at  any  one  instant  of  time  is  the  difference  of 
longitude  between  the  stations.  The  whole  object  of  the  exchange 
of  signals,  therefore,  is  to  identify  the  same  instant  of  time  on 
both  chronograph  sheets.  At  the  agreed  time  for  the  exchange 
of  signals  the  two  stations  are  thrown  into  circuit  with  the  main 
telegraph  line,  with  connections  so  arranged  that  signals  (momen- 
tary breaking  of  circuit)  sent  by  either  station  are  recorded  on 
both  chronographs.  No  signal,  however,  is  recorded  at  exactly 
the  same  instant  at  both  stations,  on  account  of  the  time  required 
for  its  passage  between  them.  The  difference  of  longitude  as 
based  on  the  signals  from  the  western  station  is  hence  too  large, 
and  that  based  on  the  eastern  station's  signals  correspondingly 
too  small.  The  mean  of  the  two  values  is  taken  as  the  true 
difference  of  longitude,  while  the  difference  of  the  two  values 
represents  double  the  time  of  signal  transmission.  In  the  Coast 
Survey  program  two  independent  sets  of  ten  pairs  of  stars 
each  are  observed  on  five  successive  nights,  the  observers  then 
exchanging  places  and  continuing  the  observations  in  the  same 
manner  for  five  more  nights.  Signals  are  exchanged  once  each 
night  at  about  the  middle  time  for  the  work  of  both  stations, 
the  western  station  sending  thirty  signals  at  intervals  of  about 
two  seconds,  followed  by  thirty  similar  signals  from  the  eastern 


ASTRONOMICAL  DETERMINATIONS  203 

station.  These  signals  were  Jprmerly  sent  by  the  chronometers, 
but  are  now  sent  by  tapping  a  break-circuit  signal  key.  The 
accuracy  attainable,  as  in  the  case  of  star  signals,  is  about  0.01 
of  a  second  of  time. 

118.  Longitude  Determinations  at  Sea.    Every  sea-going  vessel 
carries  one  or  more  chronometers,  the  error  and  rate  of  each  being 
determined  before  leaving  port,  so  that  the  Greenwich  time  of 
any  instant  is   always  very  closely   known.     The  local  time  for 
the    ship's    position    having   been   determined   for   any   instant 
(Art.  105),  and  the  corresponding  Greenwich  time  being  obtained 
from  the  chronometers,  it  is  only  necessary  to  take  the  difference 
of  these  times  to  have  the  ship's  longitude  from  Greenwich. 
The  result  thus  obtained  is  expressed  in  time  units,  but  is  readily 
converted  into  angular  units  by  multiplying  by  15  (Art.  113). 
In  case  of  failure  of  the  chronometers,  longitude  at  sea  can  still 
be  determined  in  a  number  of  ways  not  requiring  a  previous 
knowledge  of  Greenwich  time,  such  as  the  method  of  lunar  dis- 
tances (Art.  115).     Discussions  and  explanations  of  these  methods 
can  be  found  in  all  works  on  Navigation  and  Nautical  Astronomy. 
A  longitude  determination  at  sea  may  be  in  error  from  a  fraction 
of  a  mile  to  a  number  of  miles,  depending  on  the  surrounding 
circumstances. 

119.  Periodic  Changes  in  Longitude.   As  explained  in  Art.  112, 
the  poles  of  the  earth  are  not  fixed   in  position,   but  each  one 
apparently  revolves  about  a  mean  point  in  a  period  of  about  425 
days,  the   radius-vector  varying  (during  a  series   of  revolutions) 
between  about  0".16  and  0".36.     The  result  of  this  shifting  of 
the  poles  is  to  cause  the  longitude  of  any  point  to  oscillate  about 
a  mean  value,  the  amplitude  of  the  oscillation  depending  on  the 
location  of  the  point.     In  precise  longitude  work,  therefore,  the 
date  of  the  determination  is  an  essential  part  of  the  record. 

AZIMUTH 

120.  General  Principles.     By  the   azimuth  of  a  line   (or  a 
direction)  from  a  given  point  is  meant  its  angular  divergence  from 
the  meridian  at  that  point,  counting  clockwise  from  the  south 
continuously  up  to  360°.     From  any   intermediate  point  on  a 
straight  line  the  azimuths  towards  the  two  ends  always  differ  by 
exactly  180°,  so  that  in  any  case  it  is  only  necessary  to  determine 


204  GEODETIC  SURVEYING 

the  azimuth  in  one  direction.  In  passing  along  a  straight  line 
the  azimuth  varies  continuously  from  point  to  point,  unless  the 
line  be  the  equator  or  a  meridian.  The  cause  of  this  change  and 
the  methods  for  computing  it  are  explained  in  detail  in  Arts.  68 
to  73,  inclusive.  The  following  articles  are  concerned  solely 
with  the  determination  of  azimuth  (and  hence  of  the  meridian) 
at  any  one  given  point. 

Geodetic  azimuth  is  that  in  which  the  angular  divergence  from 
the  meridian  is  measured  in  a  plane  which  is  tangent  to  the 
spheroid  at  the  given  point.  Azimuth  obtained  from  observations 
on  heavenly  bodies,  or  astronomical  az  muth,  is  identical  with 
geodetic  azimuth  except  where  local  deviation  of  the  plumb  line 
(Art.  75)  exists.  The  geodetic  azimuth  of  a  line  from  a  given 
point  can  never  be  directly  observed,  nor  can  the  deviation  of 
the  plumb  line  be  found  by  direct  measurement.  If,  however, 
the  azimuth  of  a  line  from  a  given  point  be  found  by  computa- 
tion (Chapter  V)  from  the  azimuth  determinations  made  at 
various  other  triangulation  stations,  and  these  values  be  averaged 
in  with  the  observed  value,  the  result  may  be  assumed  to  be  free 
from  the  effects  of  plumb  line  deviation  and  to  represent  the  true 
geodetic  azimuth.  In  geodetic  work  geodetic  azimuth  is  always 
understood  unless  otherwise  specified. 

121.  The  Azimuth  Mark.  This  is  the  signal  which  gives  the 
direction  of  the  line  whose  azimuth  is  being  determined.  An 
azimuth  mark  should  not  be  placed  less  than  about  a  mile  from 
the  observer,  otherwise  a  change  of  focus  will  be  required  between 
the  heavenly  body  and  the  mark.  Experience  has  shown  that 
refocussing  during  an  observation  is  very  undesirable.  When 
azimuth  is  obtained  by  solar  observations  any  of  the  usual  day- 
time signals  (Art.  19)  may  be  used,  being  located  at  a  special 
azimuth  point  or  a  regular  triangulation  station  as  circumstances 
may  require.  When  azimuth  is  obtained  by  stellar  observations 
a  special  azimuth  point  is  generally  located  one  or  more  miles 
from  the  instrument.  The  azimuth  mark  should  be  mounted 
on  a  post  or  otherwise  raised  about  five  feet  above  the  ground, 
and  generally  consists  of  a  bull's-eye  lantern  enclosed  in  a  box 
or  placed  behind  a  screen,  a  small  circular  hole  being  provided 
for  the  light  to  pass  through  on  its  way  to  the  observer.  If  the 
diameter  of  the  hole  does  not  subtend  over  a  second  of  arc  (0.3 
of  an  inch  per  mile)  at  the  eye  of  the  observer,  the  light  will 


ASTKONOMICAL  DETERMINATIONS  205 

closely  resemble  a  star  in  botk  apparent  size  and  brilliancy,  which 
is  the  object  sought.  The  face  of  the  box  or  screen  is  often  painted 
with  stripes  or  other  design  so  that  it  may  also  be  observed  in  the 
daytime. 

122.  Azimuth  by  Sun  or  Star  Altitudes.  The  altitude  of  any 
heavenly  body  as  seen  by  an  observer  at  a  given  point  is  con- 
stantly changing,  each  different  altitude  corresponding  to  a  par- 
ticular azimuth  which  can  be  computed  if  the  latitude  and  longi- 
tude are  approximately  known.  For  the  degree  of  accuracy 
sought  by  this  method  it  is  sufficient  to  know  the  latitude  to  the 
nearest  minute  and  the  longitude  within  a  few  degrees.  The 
difference  in  azimuth  of  any  two  lines  from  the  same  point  is 
always  exactly  the  same  as  their  angular  divergence.  If,  therefore, 
the  horizontal  angle  between  the  azimuth  mark  and  the  given 
heavenly  body  is  measured  at  the  same  moment  that  the  altitude 
is  taken,  the  azmiuth  of  the  line  to  the  azimuth  mark  is  obtained 
by  simply  combining  the  computed  azimuth  of  the  heavenly  body 
with  this  measured  horizontal  angle.  The  observation  may  be 
made  with  a  transit  or  an  altazimuth  instrument.  The  probable 
error  of  a  single  determination  should  not  exceed  a  minute  of 
arc  with  the  ordinary  engineer's  transit,  nor  a  half  minute  with  the 
larger  instruments.  The  actual  error  may  be  larger  than  the 
probable  error  on  account  of  the  uncertainties  of  refraction. 

122a.  Making  the  Observation.  The  best  time  for  making 
an  observation  on  the  sun  is  between  about  8  and  10  o'clock  in 
the  morning  and  2  and  4  o'clock  in  the  afternoon.  The  sun  should 
not  be  observed  within  less  than  two  hours  of  the  meridian 
because  its  change  in  azimuth  is  then  so  much  more  rapid  than 
its  change  in  altitude;  nor  when  it  is  much  more  than  four  hours 
from  the  meridian  on  account  of  the  uncertain  refraction  at  low 
altitudes.  In  the  latitude  of  New  York  it  is  not  desirable  to 
observe  the  sun  for  azimuth  in  the  winter  time  because  its  dis- 
tance from  the  prime  vertical  during  suitable  hours  results  in  such 
a  rapid  movement  in  azimuth  as  compared  with  its  movement 
in  altitude.  Star  observations  may  be  made  at  any  hour  of  the 
night,  selecting  stars  which  are  about  three  hours  from  the  meridian 
and  near  the  prime  vertical,  and  hence  changing  but  slowly  in 
azimuth  as  compared  with  the  change  in  altitude.  The  observa- 
tions are  made  in  sets  of  two,  taking  one  reading  with  the  tele- 
scope direct  and  the  other  with  the  telescope  reversed,  the  mean 


206  GEODETIC  SURVEYING 

horizontal  and  the  mean  vertical  angle  constituting  the  observed 
values  for  that  set.  Several  independent  sets  should  be  taken  and 
separately  reduced,  the  mean  of  the  resulting  azimuths  being  the 
most  probable  value.  The  instrument  should  be  in  perfect 
adjustment  and  be  leveled  up  with  the  long  bubble  or  the  striding 
level,  and  should  not  be  releveled  except  at  the  beginning  of  each 
set.  The  center  of  the  sun  is -not  directly  observed,  but  the  read- 
ing is  taken  with  the  image  of  the  sun  tangent  to  the  horizontal 
and  vertical  hairs.  A  complete  set  is  made  up  as  follows:  Sight 
on  the  mark  and  read  the  horizontal  circle;  unclamp  the  upper 
motion  and  bring  the  sun's  image  tangent  to  the  horizontal  and 
vertical  hairs  in  that  quadrant  where  it  appears  by  its  own  motion 
to  approach  both  hairs;  note  the  time  to  the  nearest  minute  and 
read  both  circles;  unclamp  the  upper  motion,  invert  the  telescope, 
and  bring  the  sun's  image  tangent  in  that  quadrant  where  it  appears 
to  recede  from  both  hairs;  note  the  time  and  read  both  circles; 
unclamp  the  upper  motion,  sight  on  the  mark  and  read  the  hori- 
zontal circle.  A  star  set  is  taken  in  the  same  manner  except  that 
in  each  pointing  the  image  of  the  star  is  bisected  by  both  hairs.  If 
the  instrument  does  not  have  a  full  vertical  circle  the  telescope 
is  not  inverted  between  the  observations,  but  an  index  correction 
must  be  applied  to  the  observed  altitudes.  The  values  used  in  the 
computations  of  the  next  article  are  those  which  correspond  to 
the  center  of  the  observed  object.  If  for  any  reason  only  one 
observation  is  secured  on  the  sun,  thus  leaving  the  set  incomplete, 
the  observed  altitude  is  reduced  to  the  center  by  applying  a 
correction  for  semi-diameter,  and  the  observed  horizontal  angle 
is  reduced  to  the  center  by  applying  a  correction  found  by  divid- 
ing the  semi-diameter  by  the  cosine  of  the  altitude.  The  semi- 
diameter  is  taken  from  the  Nautical  Almanac  for  the  given 
time  and  date,  and  the  correction  is  added  or  subtracted 
in  accordance  with  the  particular  limb  of  the  sun  which  was 
observed. 

122b.  The .  Computation.  It  is  best  to  reduce  each  set  inde- 
pendently and  average  the  final  results.  The  observed  altitude 
must  first  be  reduced  to  the  true  altitude.  The  apparent  altitude 
of  all  heavenly  bodies  is  too  large  on  account  of  refraction,  the 
required  correction  being  found  in  Table  VIII.  The  apparent 
altitude  of  the  sun  is  also  too  small  on  account  of  parallax,  the 
amount  being  equal  to  8".9  multiplied  by  the  cosine  of  the 


ASTRONOMICAL  DETERMINATIONS  207 

I 

observed  altitude,  but  this  correction  is  so  small  it  would  seldom 
be  applied  in  this  method. 

In  the  polar  triangle  ZPS,  Fig.  47,  page  166,  the  three  sides  are 
known.  ZP,  the  co-latitude,  is  found  by  subtracting  the  observer's 
latitude  from  90°.  PS,  the  polar  distance  or  co-declination,  is 
found  by  subtracting  the  declination  of  the  observed  body  from 
90°.  In  the  case  of  the  sun  the  declination  is  constantly  changing 
and  must  be  taken  for  the  given  date  and  hour  (the  time  being 
always  approximately  known).  The  sun's  declination  for  Green- 
wich mean  noon  is  given  in  the  Nautical  Almanac  for  every  day 
in  the  year,  and  can  be  interpolated  for  the  Greenwich  time  of 
the  observation;  the  Greenwich  time  of  the  observation  differs 
from  the  observer's  time  by  the  difference  in  longitude  in  hours, 
remembering  that  for  points  west  of  Greenwich  the  clock  time 
is  earlier  and  vice  versa.  ZS,  the  co-altitude,  is  found  by  sub- 
tracting the  true  (reduced)  altitude  of  the  observed  body  from  90°. 
Using  the  notation  of  Fig.  47,  we  have  from  spherical  trigonometry, 

sin  d  =  cos  z  sin  (f>  +  sin  z  cos  <j>  cos  A, 
whence 

sin  d  —  cos  z  sin  6 
cos  A  =  -    — : —        —7 — -, 
sm  z  cos  <£> 


_    lcos%[z  +  (</)  +  d)]  sin  J|>  +  (0  -  £)] 

~  N/nrko  JJ*    _    fVA    -L 


which  for  logarithmic  computation  is  reduced  to  the  form 

tan  ^A 

'cos  ^\z  —  (<p  +  0;j  sin  ^[z  —  {$>  - 

The  value  of  A  thus  found  is  the  azimuth  angle  (from  north 
branch  of  meridian)  of  the  given  heavenly  body  at  the  moment 
of  observation.  If  the  observed  body  was  east  of  the  meridian 
its  azimuth  (from  the  south  point)  equals  180°  +  A;  if  west  of 
the  meridian,  180°  —  A.  The  azimuth  of  the  azimuth  mark  is 
then  found  by  combining  the  azimuth  of  the  observed  body  with 
the  corresponding  angle  between  the  azimuth  mark  and  the 
observed  body,  the  combination  being  made  by  addition  or 
subtraction  as  the  case  requires. 

123.  Azimuth  from  Observations  on  Circumpolar  Stars.  The 
simplest  and  most  accurate  method  of  determining  azimuth  is 
by  suitable  observations  on  close  circumpolar  stars,  furnishing 
any  desired  degree  of  precision  up  to  the  highest  attainable.  In 


208  GEODETIC  SURVEYING 

northern  latitudes  the  beet  available  stars  are  a  Ursae  Minoris 
(2nd  magnitude),  d  Ursse  Minoris  (4th  magnitude),  51  Cephei 
(5th  magnitude),  and  A  Ursse  Minoris  (6th  magnitude).  Of  these 
four  a  Ursae  Minoris,  commonly  known  as  Po  aris,  is  usually 
chosen  by  engineers  on  account  of  its  brightness,  the  other  three 
being  barely  visible  to  the  naked  eye.  The  four  stars  named  may 
be  identified  by  reference  to  Fig.  50,  page  191. 

Owing  to  the  rotation  of  the  earth  on  its  axis  the  azimuth 
of  any  star,  as  seen  from  a  given  point,  is  constantly  changing, 
but  the  value  of  the  azimuth  may  be  computed  for  any  given 
instant  of  time  when  the  position  of  the  observer  is  known.  The 
most  favorable  time  for  the  observation  of  a  close  circumpolar 
star  is  at  or  near  elongation  (greatest  apparent  distance  east  or 
west  of  the  meridian),  as  its  motion  in  azimuth  is  then  reduced 
to  a  minimum;  but  entirely  satisfactory  results  may  be  obtained 
from  observations  taken  at  any  time  within  about  two  hours 
either  way  from  elongation;  the  only  point  involved  is  that  time 
must  be  known  with  increasing  accuracy  the  greater  the  interval 
from  elongation,  in  order  to  secure  the  same  degree  of  precision 
in  the  azimuth  determination.  In  any  case  the  actual  observation 
consists  in  measuring  the  horizontal  angle  between  an  azimuth 
mark  and  the  given  star,  and  noting  the  time  at  which  the  star 
pointing  is  made.  The  azimuth  of  the  mark  is  then  obtained 
by  combining  the  measured  angle  (by  addition  or  subtraction 
as  the  case  requires)  with  the  computed  azimuth  of  the  star. 
The  details  of  the  observation  will  depend  on  the  instrument 
available  and  the  degree  of  precision  desired  in  the  result.  The 
instruments  used  may  be  the  ordinary  engineer's  transit,  the 
larger  transits  equipped  with  striding  levels,  the  repeating  instru- 
ment, or  the  direction  instrument.  Close  instrumental  adjust- 
ments are  necessary  for  good  work.  The  methods  ordinarily 
used  are  the  direction  method,  the  repeating  method,  and  the 
micrometric  method.  Certain  formulas  enter  more  or  less  into 
all  the  methods. 

123a.  Fundamental  Formulas.  The  following  symbols  are 
involved  in  the  formulas  as  here  given: 

A  —  azimuth  of  star  (at   any  time)  from   north  point, 

+  when  east,  --  when  west; 
Ae  —  azimuth  of  star  at  elongation; 


ASTEONOMICAL  DETERMINATIONS  209 

Ao  =  azimuth  of  star  At  mean  hour  angle  of  n  pointings; 
n  =  number  of  pointings  to  star; 

t  =  hour  angle  of  star  (at  any  time),  +  when  star  is 
west,  —  when  east,  or  may  be  counted  westward  up 
to  24  hours  or  360°; 
te  =  hour  angle  of  star  at  elongation; 
At  =  interval  of  any  one  hour  angle  from  the  mean  of 

n  given  hour  angles; 

C  =  curvature  correction  in  seconds  of  arc; 
D  =  correction  for  diurnal  aberration  in  seconds  of  arc; 
De  =  ditto  for  a  close  circumpolar  star  at  elongation; 
(f>  =  latitude,  +  when  north,  —  when  south; 
d  =  decimation  of  star,  +  when    north,  —  when  south  ; 
Am  —  azimuth   of   mark    from   north   point,  +  to   east, 

-  to  west ; 

Z  =  azimuth  of  mark  from  south  point; 
h  =  mean  altitude  of  star; 

d  =  value  of  one  division  of  bubble  tube  in  seconds; 
w,  w'j  etc.  =  readings  of  west  end  of  bubble  tube  when  sighting 

on  star; 

W  =  mean  value  of  w,  w',  etc. ; 
e,  ef,  etc.  =  readings  of  east  end  of  bubble  tube  when  sighting  on 

star; 

E  =  mean  value  of  e,  e',  etc.; 
b  =  mean  inclination  of  telescope  axis  in  seconds  when 

sighting  on  star; 
x  =  angle  correction  in  seconds  due  to  inclination  of 

telescope  axis; 
«  =  star's  right  ascension; 
S  =  sidereal  time  at  any  instant; 
Se  =  sidereal  time  of  star's  elongation. 

a.  How  angle  at  any  instant.  The  hour  angle  of  a  star 
(in  time  units)  at  any  instant  of  sidereal  time  is  given  by  the 
formula 

t  =S  -a. 

The  corresponding  value  of  t  in  angular  units  is  obtained 
(Art.  95)  by  multiplying  by  15.  The  particular  unit  in  which  t  is 
to  be  expressed  is  always  apparent  from  the  formula  in  which  it 
occurs.  If  local  mean  time  or  standard  time  is  used  it  must  be 


210  GEODETIC  SURVEYING 

reduced  to  sidereal  time  (Art.  99)  before  being  used  in  the  formula 
for*. 

b.  Hour  angle   at   elongation.     In   the  polar  triangle   ZPp, 
Fig.  47,  page  166,  p  may  be  taken  to  represent  Polaris  or  any 
other  star  at  elongation,  or  greatest  apparent  distance  from  the 
meridian  for  the  observer  whose  zenith  is  at  Z.     In  this  triangle 
the  side  PZ  is  the  observer's  co-latitude,  the  side  Pp  is  the  star's 
co-declination,  and  the  angle  ZpP  equals  90°  on  account  of  the 
tangency  at  the  point  p.     Solving^for  the  angle  ZPp,  or  the  star's 
hour  angle  at  elongation,  we  have 

tan  6 

cos  te  =  -  -  £. 
tan  d 

c.  Time  of  elongation.      Having  found   te  from  the  formula 
in  (b),  the  sidereal  time  of  elongation  is  given  by  the  formulas 

Se  =  a  +  te     (western  elongation), 
Se  =  a  —  te     (eastern  elongation). 

The  sidereal  time  thus  obtained  is  changed  to  local  mean  time  or 
standard  time  by  Art.  100  when  so  desired. 

d.  Azimuth  at  elongation.     If  the  above  triangle  (b)  be  solved 
for  the  angle  PZp,  or  the  star's  azimuth  at  elongation,  we  have 

sin  polar  distance        cos  d 
sin  Ae  =  —     —  ,    ...  —  -,  ---  =  -  r. 
cos  latitude  cos  (/> 

e.  Reduction  to  elongation.     If  the  angle  between  the  azimuth 
mark  and  a  close  circumpolar  star  is  measured  within  about 
thirty  minutes  either  way  from  elongation,  the  measured  angle 
may  be  reduced  very  nearly  to  what  it  would  have  been  if  measured 
at  elongation  by  applying  the  following  correction: 


The  quantity  (te  —  t)  is  equivalent  to  the  sidereal  time  interval 
from  elongation,  and  may  be  substituted  directly  without  com- 
puting the  hour  angle  represented  by  t.  If  the  mean  or  standard 


ASTRONOMICAL  DETERMINATIONS  211 

time  interval  is  thus_  used  the  value  which  the  formula  gives  for 
(Ae—  A)  must  be  increased  by  yj^  part  of  itself. 

/.  Azimuth  at  any  hour  angle.  If  the  star  is  observed  at  any 
other  hour  angle  than  that  which  corresponds  to  elongation,  a  polar 
triangle  will  be  formed  similar  to  ZPp,  Fig.  47,  page  166,  but 
with  all  the  angles  oblique.  In  this  case  the  azimuth  A  at  the 
given  hour  angle  t  is  given  by  the  formula, 

sin  t 

tan  A  =  — 


sin  (f>  cos  t  —  cos  (f>  tan  d 

cot  d  sec  (j>  sin  t 
1  —  cot  d  tan  <j>  cos  t 

=  —  cot  d  sec  <j)  sin  MT^rji 

in  which 

a  =  cot  d  tan  </>  cos  t. 

g.  The  curvature  correction.  If  a  series  of  observations  are 
taken  on  a  star  the  hour  angle  and  corresponding  azimuth  must 
necessarily  be  different  for  each  pointing.  The  mean  value  of 
such  azimuths  is  frequently  desired,  and  may  of  course  be  found 
by  computing  each  azimuth  separately  and  averaging  the  results. 
The  same  value,  however,  may  be  obtained  much  more  simply 
by  computing  the  azimuth  corresponding  to  the  mean  of  the 
several  hour  angles,  and  then  applying  the  so-called  curvature 
correction  to  reduce  this  result  to  the  mean  azimuth  desired. 
The  reason  that  such  a  correction  is  required  is  because  the  motion 
of  a  star  in  azimuth  is  not  uniform,  but  varies  from  zero  at  elonga- 
tion to  a  maximum  a1  culmination.  In  the  case  of  a  close  circum- 
polar  star,  and  a  series  of  observations  not  extending  over  about 
a  half  hour,  the  curvature  correction  is  given  by  the  formula 

1     2  sin2 

C  =  tan  A0- 


,    , 
n        sin  I" 

in  which  At  is  expressed  in  angular  value,  or 


C  =  tan  Ao^^sin  1"  -2 

£  n 


212  GEODETIC  SURVEYING 

in  which  At  is  expressed  in  sidereal  seconds  of  time.  If  At  is 
expressed  in  mean-time  seconds  the  value  of  C  thus  obtained 
must  be  increased  by  TSTF  part  of  itself. 


sin  1"     =  6.7367275  -  10. 


The  sign  of  the  curvature  correction  C  is  known  from  the  fact  that 
the  true  mean  azimuth  always  lies  nearer  the  meridian  than  the 
azimuth  that  corresponds  to  the  mean  hour  angle.  From  the 
nature  of  the  case  it  is  evident  that  the  several  values  of  At  in 
time  units  may  be  obtained  directly  from  the  observed  times 
(without  changing  them  to  hour  angles)  by  taking  the  differences 
between  each  observed  time  and  the  mean  of  all  the  observed  times. 
h.  Correction  for  inclination  of  telescope  axis.  If  the  axis 
of  the  telescope  is  not  horizontal  the  line  of  sight  will  not  describe 
a  vertical  plane  when  the  telescope  is  revolved  on  this  axis,  and 
hence  the  measured  angle  between  the  star  and  the  mark  will  be 
in  error  a  corresponding  amount.  The  inclination  of  the  axis 
is  found  from  the  readings  of  the  striding  level.  If  the  level  is 
reversed  but  once  the  usual  formula  is 

b  =  ^[(w  +  w')  -  (e  +  eT)]-, 

but  if  the  level  is  reversed  more  than  once  it  is  more  convenient 
to  write 


So  far  as  the  present  purpose  is  concerned  these  formulas  are 
equally  applicable  whether  the  level  is  actually  reversed  on  the 
pivots,  or  reversed  in  direction  because  the  instrument  is  turned 
through  180°.  In  one  case  the  value  obtained  is  the  actual 
average  inclination  of  the  axis,  while  in  the  other  case  it  is  the 
net  inclination.  By  the  east  or  west  end  of  the  bubble  tube  is 
meant  literally  the  end  which  happens  to  be  east  or  west  when  the 
reading  is  taken.  The  correction  required  on  account  of  the 
inclination  6,  due  to  the  altitude  of  the  star,  is 

x  =  b  tan  h. 


ASTRONOMICAL  DETERMINATIONS  213 

The  value  of  x  thus  obtained  is  to  be  subtracted  algebraically 
from  the  computed  azimuth  of  the  mark.  Ordinarily  a  similar 
correction  for  inclination  due  to  altitude  of  mark  is  not  required, 
as  the  mark  is  generally  nearly  in  the  horizon  of  the  instrument. 
If,  however,  the  angular  elevation  (+  altitude)  or  depression 
(  —  altitude)  of  the  mark  is  reasonably  large,  the  striding  level 
should  be  read  when  pointing  to  the  mark  and  a  similar  correction 
computed.  In  this  case  the  correction  is  to  be  added  algebraically 
to  the  computed  azimuth  of  the  mark. 

i.  Correction  for  diurnal  aberration.  Owing  to  the  rotation 
of  the  earth  on  its  axis  and  the  aberration  of  light  thereby  caused, 
the  apparent  position  of  any  star  is  always  more  or  less  east  of 
its  true  position,  the  amount  of  the  displacement  depending  on 
the  position  of  the  observer  and  the  position  of  the  star.  A 
corresponding  correction  is  required  for  all  azimuths  based  on 
the  measurement  of  a  horizontal  angle  between  a  mark  and  a 
star,  and  is  given  by  the  formula 


. 

cos  h 

which  for  a  close  circumpolar  star  at  elongation  reduces  to 
£>e=0".32  cos  A. 

In  obtaining  azimuth  from  a  north  circumpolar  star  it  is  evident 
that  the  azimuth  of  the  mark  (counting  clockwise  from  either 
the  north  or  south  point)  must  be  increased  by  the  amount  of 
the  above  correction. 

j.  Reduction  of  azimuth  to  south  point.  In  making  azimuth 
determinations  by  observations  on  north  circumpolar  stars  it  is 
customary  to  refer  all  results  to  the  north  point  until  the  azimuth 
of  the  mark  is  thus  expressed.  The  azimuth  of  the  mark  from  the 
south  point  is  then  given  by  the  formula 

Z  =  ISO0  +  Am, 

in  which  proper  regard  must  be  had  to  the  negative  sign  of  A  m  if 
it  is  taken  counter-clockwise. 

123b.  Approximate  Determinations.  It  is  frequently  desirable 
to  make  approximate  determinations  of  azimuth,  either  because 
the  work  in  hand  does  not  call  for  any  greater  accuracy,  or  as  a 
preliminary  to  the  more  accurate  location  of  the  meridian.  Such 


214  GEODETIC  SURVEYING 

determinations  may  be  made  by  measuring  sun  or  star  altitudes, 
as  explained  in  Art.  122,  but  observations  on  Polaris  (or  other 
circumpolar  stars)  give  more  reliable  results  without  any  increase 
in  either  field  or  office  labor.  The  ordinary  engineer's  transit 
may  be  used  for  such  work,  and  with  proper  care  will  give  correct 
results  within  the  smallest  reading  of  the  instrument.  Since 
the  observation  is  best  made  at  or  near  elongation  the  time  of 
elongation  (c,  Art.  123a)  is  computed  beforehand,  so  that  proper 
preparation  may  be  made.  Assuming  the  instrument  to  be  in 
good  adjustment  and  carefully  leveled,  the  observation  consists 
in  reading  on  the  mark  with  telescope  direct,  reading  on  the  star 
with  telescope  direct,  reading  on  the  star  with  telescope  reversed, 
and  ending  with  a  reading  on  the  mark  with  telescope  reversed. 
The  lower  motion  must  be  left  clamped  and  all  pointings  made  with 
the  upper  motion  alone.  The  instrument  must  not  be  releveled 
during  the  set.  Both  plate  verniers  should  be  read  at  each  pointing. 
The  four  pointings  should  be  made  in  close  succession,  but  with- 
out undue  haste  or  lack  of  care.  If  the  observation  is  being  made 
at  elongation  the  first  pointing  to  the  mark  is  made  a  few  minutes 
before  the  computed  time  of  elongation,  and  the  two  star  point- 
ings as  near  as  may  be  to  the  time  of  elongation.  If  time  is  not 
accurately  known  the  star  is  followed  with  the  telescope  until 
elongation  is  evidently  reached,  when  the  necessary  observations 
are  quickly  taken.  For  five  minutes  each  side  of  elongation  the 
motion  of  the  star  in  azimuth  is  scarcely  perceptible  in  an  engineer's 
transit.  If  the  observations  are  not  taken  at  elongation  time  must 
be  accurately  known  and  read  to  the  nearest  second  at  each  star 
pointing.  The  observations  having  been  completed  the  mean 
angle  between  the  mark  and  the  star  is  obtained  from  the  four 
readings  taken,  and  it  only  remains  to  compute  the  mean  azimuth 
of  the  star  to  know  the  azimuth  of  the  mark.  If  the  star  point- 
ings were  made  within  about  ten  minutes  either  way  from  elonga- 
tion the  azimuth  of  the  star  may  be  taken  as  equal  to  its  azimuth 
at  elongation  (d,  Art.  123a).  If  the  star  pointings  were  made 
within  about  a  half  hour  either  way  from  elongation  the  angle 
between  the  mark  and  the  star  may  be  reduced  to  what  it  would 
have  been  at  elongation  by  use  of  the  formula  for  reduction  to 
elongation  (e,  Art.  123a),  the  quantity  (te—  t)  being  taken  as 
the  angular  value  of  the  time  interval  between  the  time  of  elonga- 
tion and  the  average  time  of  the  star  pointings.  If  the  observa- 


ASTRONOMICAL  DETERMINATIONS  215 

tions  are  taken  over  about  a  half  hour  from  elongation  it  is  better  to 
compute  the  true  azimuth  of  the  star  for  the  average  time  of  the 
star  pointings  (/,  Art.  123a). 

123c.  The  Direction  Method.  In  this  method  the  angle 
between  the  mark  and  the  star  is  measured  with  a  direction 
instrument  (Arts.  42-47),  the  process  being  substantially  the  same 
as  there  described  for  measuring  angles  between  triangulation 
stations.  Owing  to  the  fact  that  the  star  is  in  motion  during  the 
observations,  however,  the  angle  being  measured  is  constantly 
changing,  and  the  reductions  must  be  correspondingly  modified. 
Owing  to  the  altitude  of  the  star  serious  errors  are  introduced 
by  any  lack  of  horizontality  in  the  telescope  axis,  and  a  cor- 
responding correction  must  be  made  in  accordance  with  the  read- 
ings of  the  striding  level.  If  the  mark  is  more  than  a  few  degrees 
out  of  the  horizon  a  similar  correction  will  be  required  for  the  same 
reason.  The  observations  may  be  made  at  any  hour  angle,  good 
work  requiring  time  to  be  known  to  the  nearest  second.  A  good 
program  for  one  set  is  to  read  twice  on  the  mark  with  telescope 
direct;  then  read  twice  on  the  star  with  telescope  direct,  noting 
the  exact  time  of  each  pointing  and  the  reading  of  each  end  of 
the  striding  level  at  each  pointing;  then  read  twice  on  the  star 
with  telescope  reversed,  noting  time  and  bubble  readings  as 
before;  then  read  twice  on  the  mark  with  telescope  reversed. 
The  striding  level  is  left  with  the  same  ends  on  the  same 
pivots  throughout  the  observations.  The  mean  azimuth  of  the 
star  for  the  four  pointings  is  then  found  by  computing  the 
azimuth  corresponding  to  the  average  time  of  these  pointings 
(/,  Art.  123a),  and  then  applying  the  curvature  correction 
(g,  Art.  123a).  The  apparent  azimuth  of  the  mark  is  then  found  by 
combining  the  computed  star  azimuth  with  the  mean  measured 
angle.  The  true  azimuth  of  the  mark  (as  given  by  this  set)  is 
then  found  by  applying  to  the  apparent  azimuth  the  level  cor- 
rection and  the  aberration  correction  (h  and  i,  Art.  123a),  and 
reducing  the  result  to  the  south  point  (j,  Art.  123a).  By  taking 
a  number  of  sets  each  night  for  several  nights,  and  averaging 
the  different  results,  a  very  close  determination  of  azimuth 
may  be  secured.  With  skilled  observers  the  probable  error  of  a 
single  set  should  not  exceed  about  a  half  a  second  of  arc,  and  this 
may  be  reduced  to  a  tenth  of  a  second  by  averaging  about  twenty- 
five  sets. 


216 


GEODETIC  SURVEYING 


EXAMPLE.— AZIMUTH   BY  DIRECTION  METHOD  *— RECORD 


Station:  Mount  Nebo,  Utah. 
Instrument:  20-in.  Theodolite  No.  5. 
Star:  Polaris,  near  lower  culmination. 


Date:  July  21,  1887. 

Observer:  W.  E. 

Position  X. 


Object, 

Chron. 
Time. 

Pos. 
of 
Tel. 

Mic. 

Circle  Reading. 

Levels  and 
Remarks. 

o 

/ 

Forw. 
d. 

Back, 
d. 

Mean 
d. 

Corr. 
for 
Run. 

Cor'd 
Mean 

Az.  mark 
Az.  mark 
Star 
Star 
Star 

Star 

Mean  of 
4  times 

Az.  mark 
Az.  mark 

h.    m.     s. 

15  06  47.0 
15  10  23.3 
15  15  57.8 
15  19  41.8 

D 
D 
D 
D 
R 
R 

R 
R 

A 
B 

c 

A 
B 

C 

A 
B 

C 

A 
B 
C 

A 
B 

C 

A 
B 
C 

A 
B 
C 

A 
B 

C 

140 
140 
136 
136 
316 
316 

320 
320 

53 
53 
09 
11 
13 
15 

53 
53 

14.8 
14.6 
32.3 

14.2 
13.4 
29.7 

19.8 
19.8 
49.4 
11.8 
38.6 
05.6 

24.0 
24.8 

-0.2 
-0.2 
-0.5 
+  0.3 
-0.2 
+  0.5 

-0.2 
-0.2 

19.6 
19.6 
48.9 
12.1 
38.4 
06.1 

23.8 
24.6 

W.         E. 
43.5     27.0 

53.7     17.5 

20.6 

14.7 
14.4 
32.1 

19.1 

14.2 
13.5 
30.0 

20.4 

45.3 
44.3 
60.7 

19.2 

43.0 
43.8 
59.2 

50.1 

07.0 
07.2 
22.6 

48.7 

06.5 
06.3 
21.0 

97.2     44.5 

+  52.7 

39.5     32.3 
27.4     44.6 

12.3 

41.3 
32.0 
44.0 

11.3 

40.5 
30.3 
43.7 

39.1 

09.5 
57.5 
10.5 

38.2 

08.5 
57.3 
10.0 

66.9     76.9 

15  13  12.4 

05.8 

27.0 
17.8 
29.0 

05.3 

26.0 
16.5 
27.5 

Mean  circle 
reading: 

On  star: 
136°12'26".3S 

On  mark: 
140°53'21".9() 

24.6 

28.3 
18.7 
29.7 

25.6 

23.3 

26.5 
16.7 

28.7 

24.0 

*  Abridged  from  example  in  Appendix    No.  7,  Report   for   1897-98,  U.  S.  Coast  and 
Geodetic  Survey. 


ASTRONOMICAL  DETERMINATIONS  217 

I 

AZIMUTH  BY  DIRECTION  METHOD— COMPUTATION 


Mount  Nebo,  Utah,  July,  1887.                                             0  =  39°  48'  33"  .44 

Explanation. 

Date  and  position 

July  21,  X 

July  21,  XI 

Mean  chronometer  time 

15M3m     12s.  44 

Oh55m     10s.  06 

Chronometer  correction 

-35  .40 

-34  .62 

Sidereal  time 

15  12       37  .04 

0  54       35  .44 

a  of  polaris 

1  17       58  .16 

1  17       58  .48 

t  of  polaris  (time) 

13  54       38  .88 

-0  23       23  .04 

t  of  polaris  (arc) 

208°39'     43".  20 

-5°50'     45".  60 

3  of  polaris 

88  42       06  .  13 

88  42       06  .20 

log  cot  d 

8.35532 

8.35532 

log  tan  <j> 

9.92087 

9.92087 

log  cos  t 

9.  94323  n 

9.99773 

log  a 

8.  21942  n 

8.27392 

log  cot  d 

8.355325 

8.355319 

log  sec  (j> 

0.114537 

0.114537 

log  sin  t 

9.  680917  n 

9.  007983  n 

log  1/1  -a 

9.992861 

0.008237 

log  (—tan  A) 

8.  143640  n 

7.  486076  n 

A 

+0°47'  51".  02 

+0°  10'  31".  68 

6m25s.4         81".  0 

7m08s.8       100".  3 

2  sin2  \At 

2   49  .2         15    .6 

3   23  .1         22    .5 

At  and  —  :  —  777  — 
sin  1 

2  45  .3         14    .9 

3    19  .4         21    .7 

6  29  .3         82    .6 

7   12  .4       102    .0 

194".  1 

246".  5 

48    .5 

61    .6 

1  v  2  sin2  \At 

1.68574 

1.78958 

log"n~     sinl" 

log  (curvature  correction) 

9.82938 

9.27566 

Curvature  correction 

+0".68 

+0".19 

Mean  azimuth  of  star 

+0°47'  50".  34 

+0°10'  31".  49 

Circle  reads  on  star 

136    12   26    .38 

151    14  14   .30 

Circle  reads  on  north 

135   24  36    .04 

151   03  42   .81 

Circle  reads  on  mark 

140    53  21    .90 

156   32  25    .95 

Approx.  azimuth  of  mark 

+  5  28  45   .86 

43    .14 

Level  correction 

-3    .94 

-0    .73 

Azimuth  of  mark 

5   28  41    .92 

42    .41 

218  GEODETIC  SURVEYING 

123d.  The  Repeating  Method.  In  this  method  the  angle 
between  the  mark  and  the  star  may  be  measured  with  any  of  the 
usual  engineering  transits  or  with  the  regular  geodetic  repeating 
instrument  (Arts.  38^1),  the  process  being  substantially  the  same 
as  there  described  for  measuring  angles  between  triangulation 
stations.  The  observations  and  reductions  are  best  made  as 
described  in  Arts.  40,  40a,  and  406,  ignoring  for  the  time  being 
the  fact  that  the  angle  which  is  being  repeated  is  constantly 
changing  in  value  on  account  of  the  apparent  motion  of  the  star. 
Time  must  be  correctly  known  and  noted  to  the  nearest  second 
for  each  star  pointing,  but  only  the  total  angle  readings  are  taken, 
as  with  terrestrial  angles.  The  striding  level  (if  the  instrument 
has  one)  may  be  kept  with  the  same  ends  on  the  same  pivots 
throughout  the  observations,  and  both  ends  should  be  read  imme- 
diately after  the  1st,  3d,  4th  and  6th  star  pointings  in  each  series 
of  six  pointings.  If  the  mark  is  more  than  a  few  degrees  out  of 
the  horizon  similar  readings  of  the  striding  level  are  also  required 
for  its  pointings.  The  observations  may  be  made  at  any  hour 
angle,  but  it  is  preferable  to  work  within  a  couple  of  hours  of 
elongation. 

In  making  the  reductions  the  azimuth  of  the  mark  from  the 
north  point  is  deduced  separately  from  each  series  of  six  pointings, 
applying  the  level  correction  (h,  Art.  123a)  in  each  case,  but 
omitting  the  aberration  correction.  The  two  results  obtained  from 
the  two  series  of  6  D.  and  R.  pointings  are  averaged  together  to 
obtain  the  value  of  the  determination  as  given  by  that  set.  Two 
or  more  complete  sets  may  be  taken  and  averaged  together  as 
desired.  The  true  azimuth  of  the  mark  (as  given  by  these  sets) 
is  then  found  by  applying  the  aberration  correction  (i,  Art.  123a) 
to  this  final  mean,  and  reducing  this  result  to  the  south  point 
(j,  Art.  123a).  In  reducing  each  series  of  six  pointings  the  accum- 
ulated angle  is  divided  by  six  exactly  as  if  the  star  had  remained 
entirely  stationary.  The  mean  angle  thus  obtained  is  the  same 
as  it  would  have  been  if  the  star  had  remained  all  the  time  at  the 
mean  point  of  its  six  separate  positions.  The  corresponding 
azimuth  of  this  mean  point  is  found  by  computing  the  azimuth 
for  the  mean  of  the  six  times  at  which  the  star  pointings  were 
made  (/,  Art.  123a)  and  applying  the  curvature  correction 
(0,  Art.  123a). 

The  accuracy  attainable  by  this  method  depends  on  the  char- 


ASTRONOMICAL  DETERMINATIONS 


219 


<r>  QJ. 
§  £ 


eo  co 
corn 


iF 


Oil-HCC 


i 

<M                                                                             ^ 
O                                                                     O 

1 

(MO                                                                         0 
O 

<M                                            (M                                                                         O 

\ 

00                                                                     0 
(M                                              (M                                                                             8 

a 

1 

"p 

1 

<o                          o                                            o 

s               s                        8 

'6 

- 

CO                                              CO                                                                             t>- 

O                                                            rH                                                                                                     «M 

0 

00                                                            O                                                                                                    t^ 

rH                                                        rH                                                                                              rH 

b-  C5          CO  O  O  "^           CO'*           CO          ^  OO          CO  rH  rH  CO          CO  (N          rH 

~? 
1 

1  w 

1  > 

O  *O        ^O  O  "t1  l>        COCO        CO  CO  CO  CO        t^  rt^  CO  O5        O5  CO        COCO 

rH                           rH                                                    IO                                                                                                     »O       • 

Tt*                                                                                              CO 
rH                                                                                              i—  1 
^OfN           COlMCOOO          t>-  OS          C^lOS^O          CSC^OOS          OSrH          ^   -1- 

rH                           rH           CO           rH                           rH                                                    CO 

c 

ii 
|l 

rHtNCO^lOCO                                 rH(MCO^lOCO 

g 

Q«P^P4                    tftfQQ 

i 

c 

!  • 

H    ^ 

3  fl 

3 

5 

03                                                                                                  Oi'                  03                                                                                                  CO 

OOOrHO<MiO            t^            T^OO-^         >OT^^            00 

COO1^           T—  IrHIO               rH               H/irHkO           rHrHOl               ^t1 

COCSC3           COOrH              Ttt              ^      t>-      OS          T^COOO              rH 
Tt<TttlOlO>OO               "O               OOO^HrHrH               rH 

ft                                                 a          ft                                                 a 

H/1                                             H^       IO              "^              *O                                                        ^O              *O 
rH                                                rH       rH               rH               rH                                                            rH               rH 

I 

0 

IO                                                                                                     CO 

6                                                  6 
J    5                  u                       -§3                  t> 

S     CQ                        CQ                                 CQ     S                         CO 

220 


GEODETIC  SURVEYING 


acter  of  the  instrument  with  which  the  work  is  done.  The  probable 
error  of  the  average  value  obtained  from  a  complete  double  set 
of  twenty-four  pointings  should  not  exceed  about  five  seconds 
with  a  good  engineer's  transit,  nor  a  single  second  with  a 
12-inch  repeater;  and  these  probable  errors  may  be  much  further 
reduced  by  averaging  many  determinations  together. 


AZIMUTH   BY  REPETITIONS— COMPUTATION 
KAHATCHEE,  ALA. 


Explanation. 

Date 

June  6 

June  6 

Chronometer  time 

14h54m     17s  7 

15hllm     48s  2 

Chronometer  correction  
Sidereal  time 

-31  .1 
14  53       46    6 

-31  .1 
15   11        17    1 

&                           . 

1   21       20    3 

1   21       20  .3 

Hour-angle  (t)  

13  32       26  .3 

13  49       56  .8 

t  in  arc 

203°  06'  34"  5 

207°  29'  12"  0 

log  sin  d> 

9.73876 

9  73876 

log  cos  t  

9  .  96367  n 

9.  94798  n 

log  sin  <j)  cos  t 

9  70243  n 

9  68674  n 

sin  0  cos  t  

-  0  5040 

-  0.4861 

cos  <p  tan  d  

+  38.7399 

+  38.7399 

cos  <j>  tan  d  —  sin  $  cos  t.-  
log  sin  t  
log  (cos  <£  tan  d  —  sin  <j>  cos  t)  ..  . 
log  (  —  tan  A)                     .    . 

+  39.2439 
9.  593830  n 
1.593772 
8  000058  n 

+  39.2260 
9.664211  n 
1.593574 
8.070637  n 

A  

+0°  34'  22".  7 

0°40'  26".  9 

,   2sin2^ 

7ra47*.7         119".  3 
5  09  .7           52    .3 
1  26  .7            4    .1 

7m048.2           98".  1 
4  30  .2           39    .8 
1    54  .2             7.1 

sin  1" 

1   52  .3             6    .9 
4   54  .3           47    .2 
7   37  .3         114    .0 

2  26  .8           11    .8 
4  25  .8           38    .5 
6  35  .8          85   .4 

1      2  sin2  yt 

343    .8 
57    .3 

1  .  7582 

280    .7 
46    .8 
1.6702 

gn       sinl" 

log  (curvature  correction)  
Curvature  correction       .  .    . 

9.7583 
+0  6 

9.7408 
+0  6 

Mean  azimuth  of  star  
Angle  star-mark 

+  0°34'22".l 
72    57  50     2 

+  0°  40'  26".  3 

72    51   46     7 

Level  correction  
Corrected  angle 

-   1    .6 

48     6 

-    1    .8 
44     9 

Azimuth  of  mark  E.  of  N.  .  .  .  .- 

73    32   10    .7 

73    32   11    .2 

ASTRONOMICAL  DETERMINATIONS  221 

123e.  The  Micrometric  Method.  In  this  method  the  angle 
between  the  mark  and  the  star  is  measured  with  an  eyepiece 
micrometer,  no  use  whatever  being  made  of  the  horizontal-limb 
graduations.  Any  form  of  transit  or  theodolite  may  be  used 
that  contains  an  eyepiece  micrometer  arranged  to  measure 
angles  in  the  plane  denned  by  the  optical  axis  and  the  horizontal 
axis  of  the  telescope.  An  eyepiece  micrometer  is  essentially 
the  same  as  the  micrometer  found  on  the  microscopes  of  direc- 
tion instruments  and  described  in  Art.  45.  When  the  observing 
telescope  is  fitted  with  an  eyepiece  micrometer  the  moving  hairs 
lie  in  the  focal  plane  of  the  objective  and  pass  across  the  images 
of  the  objects  viewed.  When  the  angle  between  two  objects  is 
small  (about  two  minutes  or  less)  it  may  be  assumed  with  great 
exactness  to  be  proportional  to  the  distance  between  the  corre- 
sponding images  in  the  telescope,  and  this  distance  is  measured 
by  the  micrometer  screw  with  great  precision.  In  applying  this 
method  to  the  determination  of  azimuth  the  mark  is  placed  nearly 
in  the  vertical  plane  through  the  star,  and  the  small  horizontal 
angle  between  the  mark  and  the  star  is  determined  from  measure- 
ments made  entirely  with  the  micrometer,  leaving  all  the  hori- 
zontal motions  of  the  instrument  clamped  in  a  fixed  position. 
The  azimuth  of  the  mark  is  then  obtained  by  combining  this 
angle  with  the  computed  azimuth  of  the  star. 

In  the  eyepiece  micrometer  the  value  of  the  angle  measured 
is  not  given  directly  by  the  readings  taken,  as  these  indicate 
only  the  number  of  revolutions  made  by  the  screw.  The  reading 
is  commonly  taken  to  the  nearest  thousandth  of  a  revolution, 
the  whole  number  of  revolutions  being  read  from  the  comb  scale, 
the  tenths  and  hundredths  from  the  graduations  on  the  head, 
and  the  thousandths  by  estimation.  In  order  to  convert  the  read- 
ing into  angular  value  it  is  necessary  to  know  the  angular  value 
of  one  turn  of  the  micrometer  screw.  The  value  of  one  turn  of 
the  screw  is  found  by  measuring  therewith  an  angle  whose  value 
is  already  known.  The  value  of  such  an  angle  may  be  found  by 
measuring  it  directly  with  the  horizontal  circle,  or  by  computing 
it  from  linear  measurements.  The  value  of  one  turn  of  the  screw 
may  also  be  obtained  by  observations  on  a  close  circumpolar  star 
near  culmination,  since  the  angle  between  any  two  positions  of 
the  star  is  readily  computed  from  the  times  of  observation,  and  the 
necessary  reductions  are  then  easily  made. 


222  GEODETIC  SURVEYING 

As  already  stated,  the  eyepiece  micrometer  measures  angles 
in  the  plane  defined  by  the  optical  axis  and  the  horizontal  axis 
of  the  telescope,  and  the  corresponding  horizontal  angle  must 
hence  be  obtained  by  a  suitable  reduction  for  the  given  altitude. 
To  measure  the  horizontal  angle  between  two  objects  at  different 
elevations,  therefore,  it  is  necessary  to  find  the  micrometer  value 
for  the  distance  of  each  object  from  the  line  of  collimation,  reduce 
each  value  to  the  horizontal  for  the  corresponding  altitude,  and 
combine  the  results  for  the  complete  horizontal  angle.  The  reduc- 
tion in  each  case  is  effected  by  multiplying  the  micrometer  value 
by  the  secant  of  the  altitude.  In  the  case  of  azimuth  determina- 
tions the  reduction  must  necessarily  be  made  for  the  star,  but 
need  not  be  made  for  the  mark  unless  it  is  several  degrees  out  of 
the  horizon. 

The  micrometric  method  may  be  used  at  any  hour  angle, 
but  unless  the  star  is  near  elongation  it  will  pass  out  of  the  safe 
range  of  the  micrometer  after  but  two  or  three  sets  of  observa- 
tions have  been  secured.  If  the  mark  is  placed  about  one  or 
two  minutes  nearer  the  meridian  than  the  star  at  elongation, 
the  observations  may  be  carried  on  within  an  hour  or  more  each 
way  from  elongation,  and  a  small  error  in  tune  will  have  little 
or  no  effect  on  the  result.  In  Coast  Survey  Appendix  No.  7, 
Report  for  1897-98,  the  following  procedure  is  recommended: 
"  The  micrometer  line  is  placed  nearly  in  the  line  of  collimation  of 
the  telescope,  a  pointing  made  upon  the  mark  by  turning  the 
horizontal  circle,  and  the  instrument  is  then  clamped  in  azimuth. 
The  program  is  then  to  take  five  pointings  upon  the  mark; 
direct  the  telescope  to  the  star;  place  the  striding  level  in  posi- 
tion; take  three  pointings  upon  the  star  with  chronometer  times; 
read  and  reverse  the  striding  level;  take  two  more  pointings  upon 
the  star,  noting  the  times;  read  the  striding  level.  This  com- 
pletes a  half-set.  The  horizontal  axis  of  the  telescope  is  then 
reversed  in  the  wyes;  the  telescope  pointed  approximately  to 
the  star;  the  striding  level  placed  in  position;  three  pointings 
taken  upon  the  star  with  observed  chronometer  times;  the  strid- 
ing level  is  read  and  reversed;  two  more  pointings  are  taken 
upon  the  star,  with  observed  times;  the  striding  level  is  read;  and 
finally  five  pointings  upon  the  mark  are  taken."  In  reducing 
such  a  set  of  observations  the  micrometer  reading  for  the  line  of 
collimation  is  taken  as  the  mean  of  all  the  readings  on  the  mark, 


ASTRONOMICAL  DETERMINATIONS 


223 


2  W 


3 


i  || 

pQ    -43     CP 

g  I  fi 


O       03  ^ 

T-H         [i^  G 

o^| 

»  I  J 

§-  PH 

.-    2  •• 

-p  43  f-i 

cJ      wj  o3 

•^      fl  -^ 


00 

«  1 

i 

"o       .w-  g 

1-1    "S   0       *O      °  ^   '               rf.                                                                                  » 

«!«£§§      |                                 § 

"<              "6-T-H    T-H                         S                                                                                        S 

1 

^ 

>5 

T—t    T-H                I—  I    T— 

co  co      eow 

00 

CO         CO         O  'O 
i-H          T-H          O5  b- 
CO         CO         <M  (N 

oo      oo 

O5T-H  O5 

b~  OO  l>- 
(N  <M  <M 

00 

icrometer  Res 

1 

05  00         O^t 

b~  oo      o  ^ 

CO  CO         r^  Tf 

o      ^      o  o 

CO        O        O  O 

Tt*             -~            T-H   T-H 

O  O  O 

I 

a 

0 

oo 

T—  1 

00        00 

T-H              T-H 

00 

JJ 

^ 

>O  O5         tO  <N  O5                       CO  b* 
O  'O         rJH  00(M                       05  CO 

Tfl    CO   T-H 

£ 

c 
'5 

<M 

d 
35 

T-H  00         W  CO  r-l                      CO  CO 
CO  T-H         rH 

i-H  (N 

** 

- 

ro  jrt        f^  crt  f^                     C^  T-H        t^»  b*  ^O 

00  -^         T-H  CO  00                     IO  00 
'O  O        CO  <N  ^                     C^l  ^ 

s 

CO  CO          <M  i—  1  O                        T-H  T-H 

T-H  O5rH 
<N  CO  CO 

Chronometer 
Time. 

po      ^>c 

00  (N         »O  CC 

CO  CO         O  »- 

8t^      oo  o~ 
0        OC 

05 

o              oo  t- 

QO                        ^^  ^ 
rt<                    O  <M 

O5                    <M  (N 

CO  CO  T-H 

00  CO  00 
Tfl  CO  iO 

(M  CO  CO 

CO 

CO 

CO 

1 

O5 

O5CO        <M 

OO5 

10 

g 

1 

O5  l>          t>- 

T—  I 

1  <x 

O5  O 

T-H 

K 

O5          r~l 

1    05 

8 
P4 

c 
oo     o-l- 

r 

00 

CO 
0    1 

3 

ooo      oo 

O5  t>- 

CO 

t,|;|i 

| 

H                 W             ^                 ^ 

o 

IP  1 

rH  T~!  o_ 


II    II    || 
V^VJVQ 


00  CO 

T^T^ 

o    o 

0000 
iO  ^O 


1 


**   -   M 

•s-s8 

CP    O) 

^^3 

^3T3 
11 


M    02 
*0*0 


^ 


%    ^T(H  ^ 

«*"««-»'  O 

o  Or^  ^ 

«*_«+-.  o 


224  GEODETIC  SURVEYING 

AZIMUTH  BY  MICROMETRIC  METHOD— COMPUTATION 


Collimation  reads  $(18  • 3134  + 18  • 2808)  =  18t  • 2971 

Mark  east  of  collimation,  18.3134-18.2971         =0.0163=     02". 02 

Circle  E.,  star  E.  of  collimation 

(18. 4042-18. 2971)(1. 1690)=  0  .1252 
Circle  W.,  star  E.  of  collimation 

(18. 2971-18. 0912) (1. 1695)=  0  .2408 
Mean,  star  E.  of  collimation  =  0  .1835=     22    .70 


Mark  west  of  star  =     20    . 68 

Level  correction  (1 . 55)  (0 . 92)  (0 . 606)  =  -  0    .86 


Mark  west  of  star,  corrected  =     19   . 82 


Mean  chronometer  time  of  observation  =     21h  10m  36s  .6 
Chronometer  correction  =—2     11    28    .2 

Sidereal  time  =     18    59    08    .4 

a  =        1     20    07    .4 


Hour-angle,  t,  in  time  17h  39m  01s  .0 

Hour-angle,  t,  in  arc  264°  45'   15".0 

log  cot  d  =     8.34362 

log  tan  0  =     9 . 78436 

log  cos  t  =     8 . 96108  n 


log  a  =  7. 08906  n 

log  cot  d  =  8.343618 

log  sec  0  =  0.068431 

log  sin  t  =  9. 998177  n 

log  1/1 -a  =  9.999467 


log  (-tan  A)  =     8. 409693  n 

A  =+1°  28'  16". 91 

log  12. 67  =     1.10278 


log  curvature  correction     =     9 . 51247 
Curvature  correction  —0    .33 

Diur.  aber.  corr.  =  +0    .32 


Mean  azimuth  of  star         =  + 1°  28'  16" .  90 
Mark  west  of  star  19    .82 


Azimuth  of  mark,  E.  of  N.  =  + 1°  27'  57"  .08 


ASTRONOMICAL  DETERMINATIONS  225 

and  all  micrometer  readings  am  referred  to  this  value.  Since  the 
star  is  changing  rapidly  in  altitude  the  star  micrometer  readings 
are  reduced  to  the  horizontal  for  the  mean  altitude  of  each  half- 
set,  the  altitude  of  the  star  being  occasionally  read  and  inter- 
polated for  any  desired  time.  The  mean  azimuth  of  the  star 
for  each  set  is  found  by  computing  the  azimuth  corresponding 
to  the  average  time  of  the  pointings  (/,  Art.  123a),  and  applying 
the  curvature  correction  (g,  Art.  123a).  The  apparent  azimuth 
of  the  mark  is  then  found  by  combining  the  computed  star  azimuth 
with  the  measured  angle  (reduced  to  the  horizontal).  The  true 
azimuth  of  the  mark  (as  given  by  this  set)  is  finally  found  by  apply- 
ing to  the  apparent  azimuth  the  level  correction  and  the  aberra- 
tion correction  (h  and  i,  Art.  123a),  and  reducing  the  result  to  the 
south  point  (j,  Art.  123a). 

The  time  occupied  in  taking  a  set  of  observations  in  the  man- 
ner above  specified  should  not  average  over  fifteen  minutes, 
so  that  a  number  of  sets  may  be  taken  in  a  single  night.  By 
averaging  the  results  of  a  number  of  nights'  work  a  very  close 
determination  of  azimuth  may  be  secured.  The  method  is  more 
accurate  than  the  direction  method  or  the  repeating  method. 
With  skilled  observers  the  probable  error  of  the  mean  of  25  or  30 
sets  should  be  less  than  a  tenth  of  a  second. 

124.  Azimuth  Determinations  at  Sea.  It  is  sometimes  neces- 
sary to  make  an  azimuth  determination  at  sea  in  order  to  test 
the  correctness  of  the  ship's  compasses.  The  method  commonly 
employed  is  to  measure  the  altitude  of  the  sun  or  one  of  the  brighter 
stars,  and  at  the  same  instant  take  its  bearing  as  shown  by  the 
compass  to  be  tested.  The  azimuth  of  the  given  heavenly  body 
is  then  computed  from  its  observed  altitude  and  the  result  reduced 
to  a  bearing.  The  difference  between  the  observed  bearing  and 
the  computed  bearing  is  the  error  of  the  compass.  The  method 
and  reductions  for  the  azimuth  observation  are  the  same  as 
explained  in  detail  in  Arts.  122,  122a,  and  1226,  except  that  the 
observation  consists  in  measuring  the  altitude  above  the  sea 
horizon  by  means  of  a  sextant,  and  that  a  correction  for  dip 
(Art.  105)  must  be  made.  The  latitude  and  longitude  of  the  ship's 
position  are  always  sufficiently  well  known  for  use  in  the  reduc- 
tions. The  computed  bearing  should  not  be  in  error  over  a  few 
minutes,  which  is  very  much  closer  than  it  is  possible  to  take  the 
compass  bearing. 


226  GEODETIC  SURVEYING 

125.  Periodic  Changes  in  Azimuth.  As  explained  in  Art.  112, 
the  poles  of  the  earth  are  not  fixed  in  position,  but  each  one  appar- 
ently revolves  about  a  mean  point  in  a  period  of  about  425  days, 
the  radius-vector  varying  (during  a  series  of  revolutions)  between 
about  0".16  and  0".36.  The  result  of  this  shifting  of  the  poles 
is  to  cause  the  azimuth  of  a  line  from  a  given  point  to  oscillate 
about  a  mean  value,  the  amplitude  of  the  oscillation  depending 
on  the  location  of  the  point.  In  precise  azimuth  work,  therefore, 
the  date  of  the  determination  is  an  essential  part  of  the  record. 


CHAPTER  VIII 
GEODETIC   MAP   DRAWING 

126.  General  Considerations.  The  object  of  a  geodetic  map  or 
chart  is  to  represent  on  a  flat  surface,  with  as  much  accuracy  of 
position  as  possible,  the  natural  and  the  artificial  features  of  a  given 
portion  of  the  earth's  surface.  It  is  presumed  that  the  engineer 
is  familiar  with  the  lettering  of  maps  and  the  usual  methods  of 
representing  the  natural  or  topographical  features,  and  such  mat- 
ters are  not  here  considered.  The  artificial  features  of  a  map 
are  the  meridians  and  parallels,  the  triangulation  system  or  other 
plotted  lines  of  location,  and  any  lines  which  may  be  drawn  to 
determine  latitude,  longitude,  azimuth,  angles,  distances,  or  areas. 

In  an  absolute  y  perfect  map  the  meridians  and  other  straight 
lines  (in  the  surveying  sense),  would  appear  as  straight  lines;  the 
meridians  would  show  a  proper  convergence  in  passing  towards 
the  poles ;  the  parallels  of  latitude  would  be  parallel  to  each  other 
and  properly  spaced,  and  would  cross  all  meridians  at  right  angles; 
all  points  would  be  properly  plotted  in  latitude  and  longitude;  and 
azimuths,  angles,  distances  and  areas  would  everywhere  scale 
correctly.  On  account  of  the  spheroidal  shape  of  the  earth,  it 
is  evident  that  such  a  map  is  an  impossibility,  except  for  very 
limited  areas.  Some  form  of  distortion  must  necessarily  exist 
in  any  representation  of  a  double  curved  surface  on  a  flat  sheet. 
By  selecting  a  type  of  projection  depending  on  the  use  to  be  made 
of  the  map,  however,  the  distortion  may  be  minimized  in  those 
features  where  accuracy  is  most  desired,  and  entirely  satisfactory 
maps  produced.  The  principal  types  of  map  projection,  as 
explained  in  the  following  articles,  are  the  cylindrical,  the  trape- 
zoidal, and  the  conical,  these  terms  referring  to  the  considerations 
governing  the  plotting  of  the  meridians  and  parallels. 

In  the  work  of  plane  surveying  the  areas  involved  are  usually 
of  such  small  extent  that  no  appreciable  error  is  introduced  in 
plotting  by  plane  angles  and  straight  line  distances,  drawing  all 

227 


228  GEODETIC  SUKVEYING 

meridians  or  other  north  and  south  lines  perfectly  straight  and 
parallel,  and  all  parallels  or  other  east  and  west  lines  also  straight 
and  parallel  and  at  right  angles  with  the  meridians.  On  account 
of  the  larger  areas  involved  in  geodetic  work  it  is  generally 
necessary  to  plot  the  meridians  and  parallels  first  (in  accordance 
with  the  selected  type  of  projection  and  the  scale  of  the  map), 
and  then  plot  each  fundamental  point  of  the  survey  by  means  of 
its  latitude  and  longitude  without  regard  to  angles  or  distances. 
The  smaller  details  may  then  be  plotted  as  in  plane  surveying. 
In  a  geodetic  map  thus  plotted  the  unavoidable  distortion  is 
reduced  and  distributed  as  much  as  possible. 

The  true  lengths  of  1°  of  latitude  and  longitude  at  the  latitude 
<j>  are  given  by  the  formulas 

1°  of  latitude  )  *a(\  -  e2) 


at  the  lat.  <£    J         180(1  -  e2  sin2  </>)§' 

1°  of  longitude  )    _  na  cos  <£ 

at  the  lat.  <f>    }  '  '  180(1  -  e2  sin2  <f>)i' 

in  which  formulas  the  letters  have  the  significance  and  values  of 
Arts.  67  and  69.  The  values  of  one  degree  of  latitude  and  longitude 
are  given  for  a  number  of  latitudes  in  Table  IX,  and  may  be 
interpolated  for  intermediate  latitudes. 

Since  the  radius  of  curvature  of  the  meridian  section  increases 
from  the  equator  to  the  poles  it  follows  that  the  above  formula 
for  the  length  of  a  degree  of  latitude  can  only  be  correct  in  the 
immediate  vicinity  of  the  given  latitude.  The  true  length  L 
of  a  meridian  arc  extending  from  the  equator  to  any  latitude  </> 
is  given  by  the  formula 

L  =  a(l  -  e}2(M<j)  -  N  sin  2<f>  +  P  sin  4<£  -  etc.), 
in  which 


M  =  1  +  fe2  +  £fe4  +.  .  ., 
N  =e2 


For  the  length  I  of  a  meridian  arc  from  the  latitude  cj>  to  the  lati 
tude  (j>f,  therefore,  we  have  practically 


-  sn  2<£ 

+  P(sin  40'  -  sin  40)]. 


GEODETIC  MAP  DKAWING  229 

Substituting  the  values  of  <ia  and  e  from  Art.  67,  and  reducing 
the  formula  to  its  simplest  form,  we  have 

I  =  A(<t>'  -  <f>)  -  B  sin  ($'  -  <£)  cos  (<£'  +  <£) 

+  C  sin  2  (<£'  -  <f>)  cos  2(<f>'  +  <f>)9 

in  which  (/>  and  (£>'  in  the  first  term  of  the  second  member  are  to 
be  expressed  in  degrees  and  decimals,  and  in  which 

C  metric,  111133.30  (  metric,  5.0458443 

=  (  feet,       364609.84  =  {  feet,       5.5618285 

C  metric,     32434.25  =    C  metric,  4.5110039 

=  (  feet,       106411.37  =  (  feet,       5.0269881 

(  metric,  34.41  _    (  metric,  1.5366847 

=  (  feet,  112.89  =  (  feet,       2.0526689 

127.  Cylindrical  Projections.  The  distinguishing  feature  of 
all  cylindrical  projections  consists  in  the  projection  of  the  given 
area  on  the  surface  of  a  right  cylinder  (of  special  radius)  whose 
axis  is  the  same  as  the  polar  axis  of  the  earth.  The  flat  map 
desired  is  then  produced  by  the  development  of  this  cylinder. 
In  all  forms  of  this  projection  the  meridians  are  projected  by  the 
meridional  planes  into  the  corresponding  right  line  elements  of 
the  cylinder,  so  that  after  development  the  meridians  appear  as 
equidistant  parallel  straight  lines.  The  parallels  of  latitude 
are  projected  into  the  circular  elements  of  the  cylinder  in  a  number 
of  different  ways,  but  in  any  case,  after  development,  appear  as 
parallel  straight  lines  crossing  the  meridians  everywhere  at  right 
angles.  The  three  most  common  types  of  this  projection  are 
explained  in  the  following  articles. 

127a.  Simple  Cylindrical  Projection.  In  this  type  of  pro- 
jection, as  illustrated  in  Fig.  54,  page  230,  the  cylinder  is  so  taken 
as  to  intersect  the  spheroid  at  the  middle  latitude  of  the  area  to  be 
mapped,  the  parallels  of  latitude  being  projected  into  the  cylinder 
by  lines  taken  normal  to  the  surface  of  the  spheroid.  It  is  evident 
from  the  figure  that  the  parallels  will  not  be  represented  by  equi- 
distant lines,  but  will  separate  more  and  more  in  advancing  towards 
the  poles.  This  distortion  in  latitude  is  offset  to  a  certain  extent 
by  a  similar  error  in  longitude,  caused  by  the  lack  of  convergence 
in  the  plotted  meridians,  so  that  the  various  topographical  features 
remain  approximately  true  to  shape.  On  account  of  the  varying 


230 


GEODETIC  SURVEYING 


distortion  in  both  latitude  and  longitude  no  single  scale  can  be 
correctly  applied  to  all  parts  of  such  a  map.  For  the  true  lengths 
of  one  degree  of  latitude  or  longitude  see  Table  IX  or  Art.  126. 
The  projected  distance  x  between  the  meridians,  per  degree  of 
longitude,  due  to  the  middle  latitude  </>',  is  given  by  the  formula 


no,  T 


180 


r      cos  $' 

[  (1  -  e2  sin2~57)Tj ' 


and  the  projected  distance  y,  from  the  equator  to  any  parallel 
<£,  by  the  formula 


cos 


ae2  sin 


/Midd 


leLat.0' 


Equator 


Any   Lat.£> 


FIG.  54. — Simple  Cylindrical  Projection. 

in  which  formulas  the  letters  have  the  significance  and  values 
of  Arts.  67  and  69. 

When  the  cylinder  is  taken  tangent  to  the  equator  (making 
<t>'  =  0),  the  factor  in  the  brackets  reduces  to  unity,  and  we  have 


na 
180 


and 


ae    sn 


y  =  a  tan  <j>  —  —.  --  9   .  ! 
(1  —  e2  sin 


In  making  a  map  by  this  method  the  meridians  and  parallels 
are  spaced  in  accordance  with  the  above  formulas,  and  the  funda- 
mental points  of  the  survey  are  then  plotted  by  latitudes  and 
longitudes.  For  small  areas  (10  square  miles)  within  about  45° 
of  the  equator  there  is  not  much  distortion  in  such  a  map.  The 
amount  of  the  'distortion  in  any  case  is  readily  obtained  by  com- 


GEODETIC  MAP  DK AWING 


231 


paring  the  results  given  by^the  true  formulas  and  the  formulas 
used  for  the  projection. 

127b.  Rectangular  Cylindrical  Projection.  In  this  type  of 
projection,  as  illustrated  in  Fig.  55,  the  cylinder  is  so  taken  as  to 
intersect  the  spheroid  at  the  middle  latitude  of  the  area  to  be 
mapped,  and  the  meridians  are  correctly  developed  on  the  ele- 
ments of  the  cylinder,  so  that  in  the  finished  map  the  parallels 
are  spaced  true  to  scale.  The  error  due  to  the  lack  of  convergence 
of  the  meridians  still  remains,  so  that  the  same  scale  can  not  be 
applied  to  all  parts  of  the  map.  The  distortion  in  longitude  is  more 
apparent  than  in  the  preceding  projection,  because  no  distor- 
tion exists  in  latitude.  As  in  the  previous  case  the  meridians 
are  spaced  true  to  scale  along  the  central  parallel. 


Middle  Lat.0' 


Equator 


\ 


X 

X 

X 

X 

X 

X 

FIG.  55. — Rectangular  Cylindrical  Projection. 

In  making  a  map  by  this  method  the  central  meridian  and 
parallel  are  first  drawn  and  graduated  to  scale,  using  Table  IX 
or  the  formulas  of  Art.  126.  The  remaining  parallels  and  meridians 
are  then  drawn,  and  the  survey  plotted  by  latitudes  and  long- 
itudes. For  small  areas  (10  square  miles)  within  about  45°  of 
the  equator  there  is  not  much  distortion  in  such  a  map,  straight 
lines  on  the  ground  being  straight  on  the  map,  and  angles  and 
distances  scaling  correctly.  The  plotting  for  such  an  area  may 
therefore  be  done  by  latitudes  and  longitudes,  or  by  angles  and 
distances,  as  in  plane  surveying. 

127c.  Mercator's  Cylindrical  Projection.  This  type  of  pro- 
jection, which  is  largely  used  for  nautical  maps,  is  illustrated 
in  Fig.  56,  page  232.  As  in  the  simple  cylindrical  projection, 
the  space  between  the  parallels  constantly  increases  in  advancing 
from  the  equator  towards  the  poles,  but  the  spacing  is  governed 
by  an  entirely  different  law.  In  Mercator's  cylindrical  projec- 
tion the  cylinder  is  taken  as  tangent  at  the  equator,  so  that  the 


232 


GEODETIC  SURVEYING 


spacing  of  the  meridians  along  the  equator  is  true  to  scale  in  the 
finished  map.  As  the  plotted  meridians  fail  to  converge,  the 
distance  between  them  is  too  great  at  all  other  points,  the  extent 
of  the  distortion  becoming  more  and  more  pronounced  as  the 
latitude  increases.  To  offset  this  condition  the  distance  between 
the  parallels  is  also  distorted  more  and  more  as  the  latitude 
increases,  making  the  law  of  distortion  exactly  the  same  in  both 
cases.  In  that  part  of  the  map  where  the  distance  between  the 
meridians  scales  twice  its  true  value,  for  instance,  the  distance 
between  the  parallels  should  also  scale  twice  its  true  value. 
Since  this  distortion  factor  changes  with  the  slightest  change  of 


Equator 


~r 
i 
i 
i 
i 


FIG.  56.  —  Mercator's  Cylindrical  Projection. 

latitude,  however,  it  is  evident  that  a  satisfactory  map  will  require 
the  meridian  to  be  built  up  of  a  great  many  very  small  pieces, 
each  multiplied  in  length  by  its  own  appropriate  factor.  A  per- 
fect map  on  this  basis  requires  an  infinitesimal  subdivision  of  the 
meridian,  and  a  summation  of  these  elements  by  the  methods  of 
the  integral  calculus.  Using  the  notation  and  the  formulas  of 
Arts.  67  and  69,  and  remembering  that  the  distortion  of  any 
parallel  is  inversely  proportional  to  its  radius,  we  have  for  the 
distortion  factor  s  at  any  latitude  </>, 

a  a  (1  —  e2  sin2  <ft)* 


r       N  cos  (f> 


cos  (/> 


Multiplying  the  meridian  element,  Rd<f>,  by  the  distortion  factor 
s,  we  have  for  dy,  the  projected  meridian  element, 


dy  =  s(Rd(l>)  = 


cos  </>(! 


—  2 


GEODETIC  MAP  DEAWING  233 

t 

whence,  by  integration, 

i  nonoe     Ti     A  +  sm  <A          i     A  +  e  sm  <A1 
y  =  1.1512925  a    log   -  -  -.  —  -7  1  —  e  log!  -  -  :  —  5      i 
L      \1  —  sin  <j>/  \l  —  esm  </>/_]' 

in  which  y  is  the  projected  distance  from  the  equator  to  any 
parallel  of  latitude  </>,  and  in  which  the  formula  is  adapted  to  the 
use  of  common  logarithms.  The  value  of  x  per  degree  of  longitude, 
for  the  spacing  of  the  meridians,  is  given  by  the  formula 


xa 


In  making  a  map  by  this  method  the  meridians  and  parallels 
are  spaced  in  accordance  with  the  above  formulas,  and  the  fun- 
damental points  of  the  map  are  then  plotted  by  latitudes  and 
longitudes.  It  is  evident  that  such  a  map  will  be  true  to  scale 
only  in  the  vicinity  of  the  equator,  and  that  different  scales  must 
be  used  for  every  part  of  the  map.  If  it  is  desired,  however,  to 
have  the  map  true  to  any  given  scale  along  the  central  parallel  </>', 
it  is  only  necessary  to  divide  the  above  values  of  x  and  y  by  the 
distortion  factor  sf  corresponding  to  the  latitude  <f>'  '. 

A  rhumb  line  or  loxodrome  between  any  two  points  on  a  spheroid 
is  a  spiral  line  which  crosses  all  the  intermediate  meridians  at  the 
same  angle.  Except  for  points  very  far  apart  such  a  line  is  not 
very  much  longer  than  the  corresponding  great  circle  distance. 
Great  circle  sailing  is  sometimes  practised  by  navigators,  but 
ordinarily  vessels  follow  a  rhumb  line,  keeping  the  same  course 
for  considerable  distances.  A  rhumb  line  of  any  length  or  angle 
will  always  appear  in  Mercator's  projection  as  an  absolutely 
straight  line,  crossing  the  plotted  meridians  at  exactly  the  same 
angle  as  that  at  which  the  rhumb  line  crosses  the  real  meridians. 
When  a  ship  sails  from  a  known  point  in  a  given  direction,  there- 
fore, its  path  is  plotted  on  a  Mercator  chart  by  simply  drawing  a 
straight  line  through  the  given  point  and  in  the  given  direction. 
The  distance  traveled  by  the  ship  is  plotted  in  accordance  with  the 
scale  suitable  to  the  given  part  of  the  map.  Similarly  the  proper 
course  to  sail  between  any  two  points  can  be  scaled  directly  from 
the  map  with  a  protractor.  It  is  for  these  reasons  that  this  type 
of  projection  is  so  valuable  for  nautical  purposes. 


234 


GEODETIC  SURVEYING 


128.  Trapezoidal  Projection.  In  this  type  of  projection, 
as  illustrated  in  Fig.  57,  the  meridians  and  parallels  form  a  series 
of  trapezoids.  All  the  meridians  and  parallels  are  drawn  as 
straight  lines.  The  central  meridian  is  first  drawn  and  properly 
graduated  in  degrees  or  minutes.  The  parallels  of  latitude  are 
then  drawn  through  these  points  of  division  as  parallel  lines  at 
right  angles  to  this  meridian.  Two  parallels,  at  about  one-fourth 
and  three-fourths  the  height  of  the  map,  are  then  properly  gradu- 
ated, and  the  corresponding  points  of  division  connected  by  a  series 
of  converging  straight  lines  to  represent  the  meridians.  For 
the  correct  distances  required  in  making  the  graduations  see 

Graduated   Correctly 


Graduated 


Correctly 


Graduated 


Correctly 


FIG.  57. — Trapezoidal  Projection. 

Table  IX  or  Art.  126.  From  the  nature  of  the  construction  it  is 
plain  that  the  central  meridian  is  the  only  one  which  the  parallels 
cross  at  right  angles.  The  fundamental  points  of  such  a  map 
are  plotted  by  latitudes  and  longitudes.  For  small  areas 
(25  square  miles)  the  distortion  in  distance  is  very  slight  in 
this  type  of  map. 

129.  Conical  Projections.  The  distinguishing  feature  of  the 
conical  projections  consists  in  the  projection  of  the  given  area 
on  the  surface  of  one  or  more  right  cones  (of  special  dimensions) 
whose  axes  are  the  same  as  the  polar  axis  of  the  earth.  The 
flat  map  desired  is  then  produced  by  the  development  of  the 
cone  or  cones  thus  used.  In  some  forms  of  this  projection  the 
meridians  are  projected  into  the  right  line  elements  of  the  cones, 
while  in  other  forms  a  different  plan  is  adopted ;  so  that  in  some 
forms  the  meridians  become  straight  lines  after  development, 


GEODETIC  MAP  DRAWING 


235 


while  in  other  forms  they  appear  as  curved  lines.  The  parallels 
of  latitude  are  always  projected  into  the  circular  elements  of  the 
cone  or  cones,  and  after  development  always  appear  as  circular 
arcs.  The  four  most  common  types  of  this  projection  are  explained 
in  the  following  articles. 

129a.  Simple  Conic  Projection.  In  this  type  of  projection, 
as  illustrated  in  Fig.  58,  the  projection  is  made  on  a  single  cone 
taken  tangent  to  the  spheroid  at  the  middle  latitude  of  the  area 
to  be  mapped.  The  meridians  are  projected  into  the  right  line 
elements  of  the  cone  by  the  meridional  planes,  and  appear  as 
straight  lines  after  development.  The  meridians  are  correctly 
developed  on  the  elements  of  the  cone,  so  that  the  parallels  are 
all  spaced  true  to  scale  on  the  finished  map,  The  parallels  are 


FIG.  58. — Simple  Conic  Projection. 

drawn  as  concentric  circles  from  the  center  A,  the  distance  AC 
being  the  tangent  distance  for  the  middle  latitude.  The  central 
parallel  is  graduated  true  to  scale,  and  the  meridians  are  drawn  as 
straight  lines  from  the  center  A  through  the  points  of  division. 
For  the  tangent  distance  AC  we  have,  from  Art.  69, 


AC  =  T  =  N  cot      = 


a  cot 


(1  -  e2  sin2 


The  correct  values  for  graduating  the  meridian  and  central 
parallel  may  be  taken  from  Table  IX  or  computed  by  the  formulas 
of  Art.  126. 

When  it  is  impracticable  to  draw  the  arc  EH  from  the  center 
A  it  may  be  located  by  rectangular  coordinates  from  the  point 
Cj  as  indicated  by  the  dotted  lines.  To  find  the  coordinates  of 


236  GEODETIC  SURVEYING 

any  point  H  (see  Fig.  59)  let  $  equal  the  angular  difference  of 
longitude  subtended  by  the  arc  CH  (radius  =  r),  and  d'  equal  the 
developed  angle  subtended  by  the  same  arc  CH  (radius  =N  cot.  <£). 
Then,  since  equal  lengths  of  arc  in  different  circles  subtend  angles 
inversely  as  the  radii,  we  have 

—  -         r         -  ^  cos  *ft  -     '    A 
~  :~  =      a 


gvng 

8'=  d  sm(/>', 
whence 

x  =  AH  sin  8'  =  N  cot  <j>  sin  (8  sin  <j>), 
and 

y  =  AH  vers  <T  =  2N  cot 

These  values  of  x  and  i/  are  readily  computed  by  means  of  the 
data  given  in  Table  IX.     In  this  projection  the  coordinates  of 
A  the   different   arcs  vary  directly  as  their  radii, 

so  that  the  coordinates  of  the  remaining  parallels 
may  be  found  by  a  simple  proportion.  As  a 
check  on  the  work  the  meridians  should  be 
straight  and  uniformly  spaced. 

In  making  a  map  by  this  method  the  merid- 
ians and  parallels  are  spaced  in  accordance  with 
the  above  rules,  and  the  fundamental  points  of 
the  survey  are  then  plotted  by  latitudes  and 
longitudes.  In  this  projection  the  meridians  and 
FIG.  59.  parallels  intersect  at  the  proper  angle  of  90°,  and 

the  parallels  are  properly  spaced;  but  the  spacing 
of  the  meridians  is  exaggerated  everywhere  except  along  the 
central  parallel,  and  all  areas  are  oo  large.  Such  a  map  is  satis- 
factory up  to  areas  measuring  several  hundred  miles  each  way. 

129b.  Mercator's  Conic  Projection.  In  this  type  of  pro- 
jection, as  illustrated  in  Fig.  60,  the  projection  is  made  on  a  single 
cone,  taken  so  as  to  intersect  the  spheroid  midway  between  the 
middle  parallel  and  the  extreme  parallels  of  the  area  to  be  mapped. 
The  remaining  parallels  may  be  considered  as  projected  into  the 
cone  so  that  the  spacing  along  the  line  BF  is  exactly  proportional 
to  the  true  spacing  along  the  meridian  GHK;  or  mathematically 

EC      CD  '    chord  CE 

_     =  -    __     —  _    {IT  r*       ;r^-  -  ' 

GC      CH  '  arc  CE  ' 


GEODETIC  MAP  DRAWING 


237 


After  development  the  entire  figure  is  then  proportionately 
enlarged  until  the  spacing  of  the  parallels  is  again  true  to  scale; 
following  which  the  developed  angle  and  its  subdivisions  are 
correspondingly  reduced  in  size,  in  order  to  make  the  projected 
parallels  C'C"  and  E'E"  true  to  the  same  scale.  The  distances 
B'C'  =  arc  GC,  C'D'  =  arc  CH,  etc.,  are  found  from  Art.  126  or 
Table  IX.  The  radius  A'C'  is  then  computed  from  the  formula 


A'C' 


~  &  sin2 


A'C'  +  arc  CE       cos  <£"  (1  -  e2  sin2  </>)** 

The  remaining  radii  are  found  from  A'C'  by  a  proper  combina- 
tion of  the  known  distances  along  the  line  A'F'.     The  parallel 

A  A' 


FIG.  60. — Mercator's  Conic  Projection. 

E'E"  is  then  graduated  both  ways  from  the  central  meridian  by 
means  of  the  values  found  from  Art.  126  or  Table  IX,  and  the 
meridians  are  drawn  as  straight  lines  from  the  point  A'. 

The  parallels  may  be  plotted  by  rectangular  coordinates 
when  it  is  impracticable  to  use  the  center  A',  but  the  values  given 
in  Table  IX  are  not  correct  for  this  type  of  projection.  The 
individual  angles  at  the  apex  A'  are  readily  obtained  from  the 
radius  A'E'  and  the  subdivisions  along  the  arc  E'E",  and  the 
coordinates  are  then  found  for  this  arc  and  proportioned  for  the 
other  arcs  directly  as  their  radii. 

In  making  a  map  by  this  method  the  meridians  and  parallels 
are  drawn  in  accordance  with  the  above  rules,  and  the  fundamental 


238 


GEODETIC  SURVEYING 


points  of  the  survey  are  then  plotted  by  latitudes  and  longitudes. 
In  this  projection  the  meridians  are  straight  lines,  the  meridians 
and  parallels  cross  at  the  proper  angle  of  90°,  and  the  parallels 
of  latitude  are  properly  spaced.  The  meridians  are  properly 
spaced  on  the  parallels  C'C"  and  E'E",  but  are  a  little  too  widely 
spaced  outside  of  these  parallels,  and  a  little  too  closely  spaced 
within  these  parallels.  Areas  outside  of  these  same  parallels  are 
too  large,  while  areas  within  them  are  too  small;  but  the  total 
area  is  nearly  correct.  Mercator's  conic  projection  is  suitable 
for  very  large  areas,  having  been  used  for  whole  continents.  It 
has  also  been  largely  used  for  the  maps  in  atlases  and  geographies. 
129c.  Bonne's  Conic  Projection.  In  this  type  of  projection, 
as  illustrated  in  Fig.  61,  the  projection  is  made  on  a  single  cone 


FIG.  61. — Bonne's  Conic  Projection. 

taken  tangent  to  the  spheroid  at  the  middle  latitude  of  the  area 
to  be  mapped.  The  central  meridian  is  projected  into  the  straight 
line  AF,  with  the  parallels  spaced  true  to  scale  and  drawn  as 
concentric  circles,  in  accordance  with  the  rules  and  formulas  for 
simple  conic  projection  (Art.  129a).  Each  parallel  is  then  gradu- 
ated true  to  scale  (see  Art.  126  or  Table  IX),  and  the  merMians 
are  drawn  as  curved  lines  through  corresponding  divisions  of  the 
parallels. 

In  making  a  map  by  this  method  the  fundamental  points  of 
the  survey  must  be  plotted  by  latitudes  and  longitudes.  In  this 
projection  the  meridians  and  parallels  fail  to  cross  at  right  angles, 


GEODETIC  MAP  DRAWING 


239 


but  the  same  scale  holds  good  for  all  the  meridians  and  all  the 
parallels.  Bonne's  conic  projection  is  suitable  for  very  large 
areas,  having  been  used  for  whole  continents.  It  has  also  been 
largely  used  for  the  maps  in  atlases  and  geographies. 

129d.  Polyconic  Projection.  In  this  type  of  projection,  as 
illustrated  in  Fig.  62,  a  separate  tangent  cone  is  taken  for  each 
parallel  of  latitude,  and  made  tangent  to  the  spheroid  at  that 
parallel.  Each  parallel  on  the  map  results  from  the  development 
of  its  own  special  cone,  appearing  as  the  arc  of  a  circle  with  a 
radius  equal  to  the  corresponding  tangent  distance.  The  parallel 
A  A 


FIG.  62. — Polyconic  Projection. 

through  the  point  G,  for  instance,  is  drawn  as  a  circular  arc  with 
a  radius  equal  to  the  tangent  distance  BG,  and  so  on.  The 
central  meridian  is  drawn  as  a  straight  line,  on  which  all  the 
parallels  are  spaced  true  to  scale,  so  that  the  division  EF  equals 
the  arc  EF,  the  division  FG  equals  the  arc  FG,  and  so  on.  The  arcs 
representing  the  various  parallels  are  then  drawn  through  these 
division  points  with  the  appropriate  radii,  and  with  the  centers 
located  on  the  central  meridian.  Each  parallel  as  thus  represented 
is  then  graduated  true  to  scale,  and  the  meridians  are  drawn  as 
curved  lines  connecting  the  corresponding  divisions. 

In  making  a  map  by  this  method  the  meridians  and  parallels 
are  plotted  in  accordance  with  the  data  given  in  Table  IX,  or 


240  GEODETIC  SURVEYING 

from  corresponding  values  computed  by  the  rules  and  formulas 
of  Arts.  126  and  129a,  remembering  that  each  parallel  is  here 
equivalent  to  the  central  parallel  of  the  simple  conic  projection. 
The  plotting  is  customarily  done  by  rectangular  coordinates, 
the  meridians  and  parallels  being  taken  so  close  together  that  the 
intersection  points  may  be  connected  by  straight  lines.  The 
fundamental  points  of  the  survey  are  then  plotted  by  latitudes 
and  longitudes. 

This  type  of  projection  is  suitable  for  very  large  areas.  The 
meridians  are  spaced  true  to  scale  throughout  the  map  and  cross 
the  parallels  nearly  at  right  angles.  The  parallels  are  spaced 
true  to  scale  only  along  the  central  meridian,  and  diverge  more 
and  mpre  from  each  other  as  the  distance  from  the  central  merid- 
ian increases.  The  whole  of  North  America,  however,  may  be 
represented  without  material  distortion.  The  U.  S.  Coast  and 
Geodetic  Survey  and  the  U.  S.  Geological  Survey  have  adopted 
the  poly  conic  system  of  projection  to  the  exclusion  of  all  others. 
For  further  information  on  this  subject  see  "  Tables  for  the 
Projection  of  Maps,  Based  upon  the  Polyconic  Projection  of 
Clarke's  Spheroid  of  1866,  and  computed  from  the  Equator  to 
the  Poles;  Special  Publication  No.  5,  U.  S.  Coast  and  Geodetic 
Survey,  U.  S.  Government  Printing  Office,  1900." 

The  above  type  of  poly  conic  projection  is  sometimes  called 
the  simple  polyconic,  to  distinguish  it  from  the  rectangular  poly- 
conic,  in  which  the  scales  along  the  parallels  are  so  taken  as  to 
make  all  the  meridians  and  parallels  cross  at  right  angles.  When 
not  otherwise  specified  the  simple  polyconic  is  in  general  under- 
stood to  be  the  one  intended. 


PART  II 

ADJUSTMENT   OF   OBSERVATIONS   BY  THE 
METHOD   OF   LEAST   SQUARES 


CHAPTER  IX 

DEFINITIONS  AND  PRINCIPLES 

130.  General    Considerations.     In    various    departments    of 
science,  such  as  Astronomy,  Geodesy,  Chemistry,  Physics,  etc., 
numerous  values  have  to  be  determined  either  directly  or  indirectly 
by  some  process  of  measurement.     When  any  fixed  magnitude, 
however,  is  measured  a  number  of  times  under  the  same  apparent 
conditions,  and  with  equal  care,  the  results  are  always  found  to 
disagree  more  or  less  amongst  themselves.     With  skillful  observers, 
and  refined  methods  and  instruments,  the  absolute  values  of 
the  discrepancies  are  decreased,  but  the  relative  disagreement 
often  becomes  more  pronounced.     The  conclusion  is  obviously 
reached  that  all  measurements  are  affected  by  certain  small  and 
unknown  errors  that  can  neither  be  foreseen  nor  avoided.     The 
object  of  the  method  of  Least  Squares  is  to  find  the  most  probable 
values  of  unknown  quantities  from  the  results  of  observation, 
and  to  gage  the  precision  of  the  observed  and  reduced  values. 

131.  Classification  of  Quantities.     The   quantities   observed 
are  either  independent  or  conditioned. 

An  independent  quantity  is  one  whose  value  is  independent  of 
the  values  of  any  of  the  associated  quantities,  or  which  may  be 
so  considered  during  a  particular  discussion.  Thus  in  the  case 
of  level  work  the  elevation  of  any  individual  bench  mark  is  an 
independent  quantity,  since  it  bears  no  necessary  relation  to  the 
elevation  of  any  other  bench  mark.  While  in  the  case  of  a  triangle 

241 


242  GEODETIC  SURVEYING 

we  may  consider  any  two  of  the  angles  as  independent  quantities 
in  any  discussion  in  which  the  remaining  angle  is  made  to  depend 
on  these  two. 

A  conditioned  quantity  (or  dependent  quantity)  is  one  whose 
value  bears  some  necessary  relation  to  one  or  more  associated 
quantities.  In  any  case  of  conditioned  quantities  we  may  regard 
these  quantities  as  being  mutually  dependent  on  each  other,  or 
any  number  of  them  as  being  dependent  on  the  remaining  ones. 
Thus  if  the  angles  of  a  triangle  are  denoted  by  x,  y,  and  z,  we 
may  write  the  conditional  equation 

x  +  y  +  z  =  180°, 

and  regard  each  angle  as  a  conditioned  quantity;  or  we  may  write, 
for  instance, 

z  =  180°  -  x  -  y, 

and  regard  z  as  conditioned  and  x  and  y  as  independent. 

132.  Classification  of  Values.  In  considering  the  value  of 
any  quantity  it  is  necessary  to  distinguish  between  the  true  value, 
the  observed  value,  and  the  most  probable  value. 

The  true  value  of  a  quantity  is,  as  its  name  implies,  that  value 
which  is  absolutely  free  of  all  error.  Since  (Art.  130)  all  measure- 
ments are  subject  to  certain  unknown  errors,  it  follows  that  the 
true  value  of  a  quantity  may  never  be  known  with  absolute  pre- 
cision. In  any  case  such  a  value  would  seldom  be  any  exact 
number  of  units,  but  could  only  be  expressed  as  an  unending 
decimal. 

The  observed  value  of  a  quantity  is  technically  understood  to 
mean  the  value  which  results  from  an  observation  when  correc- 
tions have  been  applied  for  all  known  errors.  Thus  in  measuring 
a  horizontal  angle  with  a  sextant  the  vernier  reading  must  be 
corrected  for  the  index  error  to  obtain  the  observed  value  of  the 
angle;  in  measuring  a  base  line  with  a  steel  tape  the  corrections 
for  horizontal  and  vertical  alignment,  pull,  ag,  temperature,  and 
absolute  length,  are  understood  to  have  been  applied;  and  so  on. 

The  most  probable  value  of  a  quantity  is  that  value  which  is 
most  likely  to  be  the  true  value  in  view  of  all  the  measurements 
on  which  it  is  based.  The  most  probable  value  in  any  case  is 
not  supposed  to  be  the  same  as  the  true  value,  but  only  that  value 
which  is  more  likely  to  be  the  true  value  than  any  other  single 
value  that  might  be  proposed. 


DEFINITIONS  AND  PRINCIPLES  243 

133.  Observed  Values  an4  Weights.     The  observations  which 
are  made  on  unknown  quantities  may  be  direct  or  indirect,  and 
in  either  case  of  equal  or  of  unequal  weight. 

A  direct  observation  is  one  that  is  made  directly  on  the  quan- 
tity whose  value  is  desired.  Thus  a  single  measurement  of  an 
angle  is  a  direct  observation. 

An  indirect  observation  is  one  that  is  made  on  some  function 
of  one  or  more  unknown  quantities.  Thus  the  measurement 
of  an  angle  by  repetition  represents  an  indirect  observation, 
since  some  multiple  of  the  angle  is  measured  instead  of  the  single 
value.  So  also  in  ordinary  leveling  the  observations  are  indirect, 
since  they  represent  the  difference  of  elevation  from  point  to 
point  instead  of  the  elevations  of  the  different  points. 

By  the  weight  of  an  observation  is  meant  its  relative  worth. 
When  observations  are  made  on  any  magnitude  with  all  the  con- 
ditions remaining  the  same,  so  that  all  the  results  obtained  may 
be  regarded  as  equally  reliable,  the  observations  are  said  to  be  of 
equal  weight  or  precision,  or  of  unit  weight.  When  the  condi- 
tions vary,  so  that  the  results  obtained  are  not  regarded  as  equally 
reliable,  the  observations  are  said  to  be  of  unequal  weight  or  pre- 
cision. It  has  been  agreed  by  mathematicians  that  the  most 
probable  value  of  a  quantity  that  can  be  deduced  from  two  obser- 
vations of  unit  weight  shall  be  assigned  a  weight  of  two,  from  three 
such  observations  a  weight  of  three,  and  so  on.  Hence  when  an 
observation  is  made  under  such  favorable  circumstances  that  the 
result  obtained  is  thought  to  be  as  reliable  as  the  most  probable 
value  due  to  two  observations  which  would  be  considered  of  unit 
weight,  we  may  arbitrarily  assign  a  weight  of  two  to  such  an 
observation;  and  so  on.  As  the  weights  applied  in  any  set  of 
observations  are  purely  relative,  their  meaning  will  not  be  changed 
by  multiplying  or  dividing  them  all  by  the  same  number.  The 
elementary  conception  of  weight  is  therefore  extended  to  include 
decimals  and  fractions  as  well  as  integers,  since  any  set  of  weights 
could  be  reduced  to  integers  by  the  use  of  a  suitable  factor. 

134.  Most  Probable  Values  and  Weights.     In   any  set   of 
observations  the  most  probable  value  of  the  unknown  quantity 
will  evidently  be  some  intermediate  or  mean  value.     There  are 
many  types  of  mean  value,  but  manifestly  they  are  all  subject 
to  the  fundamental  condition  that  in  the  case  of  equal  values  the 
mean  value  must  be  that  common  value.     Three  of  the  common 


244  GEODETIC  SURVEYING 

types  of  mean  value  are  the  arithmetic  mean,  the  geometric  mean, 
and  the  quadratic  mean.  If  there  are  n  quantities  whose  respective 
values  are  MI,  M2)  etc.,  we  have, 

v  7i/r 

-  =  the  arithmetic  mean; 
n 


.  .  .  Mn  =  the  geometric   mean;  \  '    t     ^     (i) 
=  the  quadratic  mean; 


n 


all  of  which  satisfy  the  fundamental  condition  of  a  mean  value. 
In  the  case  of  direct  observations  of  equal  weight  it  has  been 
universally  agreed  that  the  arithmetic  mean  is  the  most  probable 
value.  In  accordance  with  this  principle;  and  the  definition  of 
weight  as  given  in  Art.  133,  it  is  evident  that  the  weight  of  the 
arithmetic  mean  is  equal  -to  the  number  of  observations.  Sim- 
ilarly, an  observation  to  which  a  weight  of  two  has  been  assigned 
may  be  regarded  as  the  arithmetic  mean  of  two  component  obser- 
vations of  unit  weight,  and  so  on,  provided  no  special  assumption 
is  made  regarding  the  relative  values  of  these  components. 
For  direct  observations  of  unequal  weight,  therefore, 

Let  z  =  the  most  probable  value  of  a  given  magnitude; 

MI,  M2,  etc.  =  the  values  of  the  several  measurements; 

Pi,  P2,  etc.  =  the  respective  weights  of  these  measurements; 
api,  ap2,  etc.  =  the  corresponding  integral  weights   due   to     the 

use  of  the  factor  a; 

m\,  nil",  etc.  =  the  ap\  unit  weight  components  of  MI  when  con- 
sidered as  an  arithmetic  mean 
mz,  m2" ',  etc.  =  similarly  for  M2,  and  so  on; 

then  we  may  write  as  equivalent  expressions 


„,        m2    +  m2    .  .  . 
M2  =  -  —  ,  etc; 

ap2  ap2 

whence 


ap2M2f  etc.; 


DEFINITIONS  AND  PRINCIPLES  245 

and,  since  the  various  values  $f  m  are  of  unit  weight, 

+  Sm2  .  .  . 


api  +  ap2  .  .  . 
or 

=  2m        2(qp.M)         2 

~  ' 


from  which  we  have  the  general  principle  : 

In  the  case  of  direct  observations  of  unequal  weight  the  most 
probable  value  is  found  by  multiplying  each  observation  by  its  weight 
and  dividing  the  sum  of  these  products  by  the  sum  of  the  weights. 
The  result  thus  obtained  is  called  the  weighted  arithmetic  mean. 
In  the  above  discussion  the  value  of  z  is  found  by  taking  the 
arithmetic  mean  of  2ap  quantities  whose  sum  is  2m,  so  that  the 
integral  weight  of  z  is  2ap.  Dividing  by  a  in  order  to  express  this 
result  in  accordance  with  the  original  scale  of  weights,  we  have 

Weight  of  z  =  2p;      ......     (3) 

or,  expressed  in  words,  the  weight  of  the  weighted  arithmetic 
mean  is  equal  to  the  sum  of  the  individual  weights. 

135.  True  and  Residual  Errors.  It  is  necessary  to  distin- 
guish between  true  errors  and  residual  errors. 

A  true  error,  as  its  name  implies,  is  the  amount  by  which  any 
proposed  value  of  a  quantity  differs  from  its  true  value.  True 
errors  are  generally  considered  as  positive  when  the  proposed 
value  is  in  excess  and  vice  versa.  Since  (Art.  132)  the  true  value 
of  a  quantity  can  never  be  known,  it  follows  that  the  true  error 
is  likewise  beyond  determination. 

A  residual  error  is  the  difference  between  any  observed  value 
of  a  quantity  and  its  most  probable  value,  in  the  same  set  of 
observations.  The  subtraction  is  taken  algebraically  in  which- 
ever way  is  most  convenient  in  the  given  discussion.  In  the  case 
of  indirect  observations  the  most  probable  value  of  the  observed 
quantity  is  found  by  substituting  the  most  probable  values  of  the 
individual  unknowns  in  the  given  observation  equation  (Art.  158). 
Residual  errors  are  frequently  called  simply  residuals. 

In  the  case  of  the  arithmetic  mean  the  sum  of  the  residual  errors 
is  zero.  This  is  proved  as  follows: 


246 


GEODETIC  SURVEYING 


Let  n  =  the  number  of  observations; 

MI,  M2,  .  : .  Mn  =  the  observed  values; 
z  =  the  arithmetic  mean; 
Vi,  v2)  .  .  .  vn=  the  residual  errors; 


then 


but 

or 

from  which 

whence 


=    2  -  Ml 

=    z  —  M  2 

Z    -  Mn 

=  nz  — 

_  2M 
n  ' 

=  SM. 


nz  - 


0; 


2v  =  0, (4) 

which  was  to  be  proved. 

In  the  case  of  the  weighted  arithmetic  mean  the  sum  of  the  weighted 
residuals  equals  zero.     This  is  proved  as  follows : 

Let  n  =  the  number  of  observations; 

MI,  M2,  .  .  .  Mn  =  the  observed  values; 

Pi>  P2,  •  •  •  Pn  =  the  corresponding  weights; 

z  =  the  weighted  arithmetic  mean; 
v\,  v2,  .  .  .  vn  =  the  residual  errors; 


then 


pi(z  - 


—  I,pM; 


but 


or 


ZpM, 


DEFINITIONS  AND  PRINCIPLES  247 

from  which  & 


=  0; 

whence 


0,     ..........     (5) 

which  was  to  be  proved. 

136.  Sources  of  Error.     The  errors  existing  in  observed  values 
may  be  due  to  mistakes,  systematic  errors,  accidental  errors,  or 
the  least  count  of  the  instrument. 

A  mistake  is,  as  its  name  implies,  an  error  in  reading  or  record- 
ing a  result,  and  is  not  supposed  to  have  escaped  detection  and 
correction. 

A  systematic  error  is  one  that  follows  some  definite  law,  and  is 
hence  free  from  any  element  of  chance.  Errors  of  this  kind  may 
be  classed  as  atmospheric  errors,  such  as  the  effect  of  refraction 
on  a  vertical  angle,  or  the  effect  of  temperature  on  a  steel  tape; 
instrumental  errors,  such  as  those  due  to  index  errors  or  imperfect 
adjustments;  and  personal  errors,  such  as  individual  peculiarities 
in  always  reading  a  scale  a  little  too  small,  or  in  recording  a  star 
transit  a  little  too  late.  Systematic  errors  usually  affect  all  the 
observations  in  the  same  manner,  and  thus  tend  to  escape  detec- 
tion by  failing  to  appear  as  discrepancies.  Such  errors,  however, 
are  in  general  well  understood,  and  are  supposed  to  be  eliminated 
by  the  method  of  observing  or  by  subsequent  reduction. 

An  accidental  error  is  one  that  happens  purely  as  a  matter  of 
chance,  and  not  in  obedience  to  any  fixed  law.  Thus,  for  instance, 
in  bisecting  a  target  an  observer  will  sometimes  err  a  little  to 
the  right,  and  sometimes  a  little  to  the  left,  without  any  assignable 
cause;  a  steel  tape  will  be  slightly  lengthened  or  shortened  by  a 
momentary  change  of  temperature  due  to  a  passing  current  of 
air,  and  so  on. 

An  error  due  to  the  least  count  of  the  instrument  is  one  that  is 
caused  by  a  measurement  that  is  not  capable  of  exact  expression 
in  terms  of  the  least  count.  Thus  an  angle  may  be  read  to  the 
nearest  second  by  an  instrument  which  has  a  least  count  of  this 
value,  but  the  true  value  of  the  angle  may  differ  from  this  reading 
by  some  fraction  of  a  second  which  can  not  be  read. 

137.  Nature  of  Accidental  Errors.     Errors  of  this  kind   are 
due  to  the  limitations  of  the  instruments  used;    the  estimations 
required  in  making  bisections,  scale  readings,  etc.,    and  the  con- 


248  GEODETIC  SURVEYING 

stantly  changing  conditions  during  the  progress  of  an  observa- 
tion. Each  individual  error  is  usually  very  minute,  but  the 
possible  number  of  such  errors  that  may  occur  in  any  one  measure- 
ment is  almost  without  limit.  In  general  it  may  be  said  that  any 
single  observation  is  affected  by  a  very  large  number  of  such  errors, 
the  total  accidental  error  being  due  to  the  algebraic  sum  of  these 
small  individual  errors.  Thus  in  measuring  a  horizontal  angle 
with  a  transit  the  instrument  is  seldom  in  a  perfectly  stable  posi- 
tion; the  leveling  is  not  perfect;  the  lines  and  levels  of  the  instru- 
ment are  affected  by  the  wind  and  varying  temperatures;  the 
graduations  are  not  perfect;  the  reading  is  affected  by  the  judg- 
ment of  the  observer;  the  target  is  bisected  only  by  estimation; 
the  line  of  sight  is  subject  to  irregular  sidewise  refraction  due  to 
changing  air  currents;  and  so  on.  As  long  as  the  component 
errors  are  all  accidental,  however,  the  total  error  may  be  regarded 
as  a  single  accidental  error. 

138.  The  Laws  of  Chance.  The  errors  remaining  in  observed 
values  after  all  possible  corrections  have  been  made  are  presumed 
to  be  accidental  errors,  and  must,  therefore  be  assumed  to  have 
occurred  in  accordance  with  the  laws  of  chance.  By  the  laws  of 
chance  are  meant  those  laws  which  determine  the  probability  of 
occurrence  of  events  which  happen  by  chance. 

By  the  probability  of  an  event  is  meant  the  relative  frequency 
of  its  occurrence.  It  is  not  only  a  reasonable  assumption  but  also 
a  matter  of  common  experience,  that  in  the  long  run  the  relative 
frequency  with  which  a  proposed  event  occurs  will  closely  approach 
the  relative  possibilities  of  the  case.  Thus  in  tossing  a  coin 
heads  may  come  up  as  one  possibility  out  of  the  two  possibilities 
of  heads  or  tails,  so  that  the  probability  of  a  head  coming  up  is 
one-half;  and  in  a  very  large  number  of  trials  the  occurrence  of 
heads  will  closely  approximate  one-half  the  total  number  of  trials. 
Probabilities  are  therefore  represented  by  fractions  ranging  in 
value  from  zero  to  unity,  in  which  zero  represents  impossibility  of 
occurrence,  while  unity  represents  certainty  of  occurrence. 

The  three  fundamental  laws  of  chance  are  those  relating  to 
simple  events,  compound  events,  and  concurrent  events. 

139.  A  Simple  Event  is  one  involving  a  single  condition  which 
must  be  satisfied.  The  probability  of  a  simple  event  is  equal  to 
the  relative  possibility  of  its  occurrence.  Thus  the  probability  of 
drawing  an  ace  from  a  pack  of  cards  is  is,  since  there  are  four 


DEFINITIONS  AND  PRINCIPLES  249 

i 

such  possibilities  out  of  52,"' and  -£%  =  TV;  but  the  probability 
of  drawing  an  ace  of  clubs,  for  instance,  is  only  fa,  since  there 
is  only  one  such  possibility  out  of  52. 

140.  A  Compound  Event  is  one  involving  two  or  more  con- 
ditions of  which  only  one  is  required  to  be  satisfied.     The  proba- 
bility of  a  compound  event  is  equal  to  the  sum  of  the  probabilities  of 
the  component  simple  events.     This  law  is  evidently  true,  since  the 
number  of  favorable  possibilities  for  the  compound  event  equals 
the  sum  of  the  corresponding  simple  possibilities,  and  the  total 
number  of  possibilities  remains  unchanged.     Thus  the  probability 
of  getting  either  a  club  or  a  spade  in  a  single  draw  from  a  pack 
of  cards  is  one-half,  because  the  probability  of  getting  a  club  is  one- 
quarter,  and  the  probability  of  getting  a  spade  is  one-quarter,  and 
}  +  i  —  i;    or  in  other  words  the  13  chances  for  getting  a  club 
are  added  to  the  13  chances  for  getting  a  spade,  making  26  favor- 
able possibilities  out  of  a  total  of  52.     The  probability  of  draw- 
ing either  a  club,  spade,  heart,  or  diamond,  equals  J  +  i'  +  i  +  }, 
which  equals  unity,  since  the  proposed  event  is  a  certainty. 

141.  A  Concurrent  Event  is  one  involving  two  or  more  con- 
ditions, all  of  which  are  required  to  be  satisfied  together.     The 
probability  of  a  concurrent  event  is  equal  to  the  product  of  the  prob- 
abilities of  the  component  simple  events.     This  law  is  evidently 
true,  since  the  number  of  favorable  possibilities  for  the  concurrent 
event  is  equal  to  the  product  of  the  corresponding  simple  pos- 
sibilities; while  the  total  number  of  possibilities  is  equal  to  the 
product  of  the  corresponding  totals   for   the  component  simple 
events.     Thus  the  probability  of  cutting  an  ace  in  a  pack  of  cards 
is  -fa,  so  that  the  probability  of  getting  two  aces  by  cutting  two 
packs  of  cards  is  -fa  X  -fa  =  gVxl^i  =  T?  X  yV  =  TBT-     It  is  evi- 
dent that  the  required  condition  will  be  satisfied  if  any  one  of  the 
four  aces  in  one  pack  is  matched  with  any  one  of  the  four  aces  in  the 
other  pack,  so  that  there  are  4X4  favorable  possibilities.     Also 
the  cutting  may  result  in  getting  any  one  of  52  cards  in  one  pack 
against  any  one  of  52  cards  in  the  other  pack,  so  that  there  are 
52X52   total   possibilities.     Multiplying   the   two   probabilities, 
therefore,  gives  the  relative  possibility  and  therefore  the  required 
probability  for  the  given  concurrent  event.     Similarly  the  propo- 
sition may  be   proved  for   a   concurrent   event   involving   any 
number   of   simple   events.     Thus   in   throwing   three   dice   the 
probability  of  getting  3  fours,  for  instance,  will  be  |X  JXf  =sir; 


250  GEODETIC  SURVEYING 

the  probability  of  drawing  a  deuce  from  a  pack  of  cards  at 
the  same  time  that  an  ace  is  thrown  with  a  die,  will  be 
iV  X  i  =  T¥  ;  and  so  on. 

In  figuring  the  probability  of  a  concurrent  event  it  is  neces- 
sary to  guard  against  two  possible  sources  of  error.  In  the 
first  place  the  probabilities  of  the  simple  events  involved  in  a 
concurrent  event  may  be  changed  by  the  concurrent  condition. 
Thus  the  probability  of  drawing  a  red  card  from  a  pack  is  f  f, 
but  the  probability  of  drawing  two  red  cards  in  succession  from 
a  pack  is  not  ff Xf|,  but  If  Xff,  since  the  drawing  of  the  first 
card  changes  the  conditions  under  which  the  second  card  is  drawn. 
In  the  second  place,  the  probability  of  a  concurrent  event  may 
be  modified  by  the  sense  in  which  the  order  of  simple  events 
may  be  involved.  Thus  in  cutting  two  packs  of  cards  the  prob- 
ability that  the  first  pack  will  cut  an  ace  and  the  second  a  king 
is  T^XT^TO;  but  the  probability  that  the  first  pack  will  cut 
a  king  and  the  second  an  ace  is  also  rVXT^rrs";  so  that  the 
probability  of  cutting  an  ace  and  a  king  without  regard  to 
specific  packs  becomes  yf ^,  and  not  yi^>  as  might  be  inferred. 

142.  Misapplication  of  the  Laws  of  Chance.  The  probability 
of  a  given  event  is  the  relative  frequency  of  its  occurrence  in 
the  long  run,  and  not  in  a  limited  number  of  cases.  It  is  not 
to  be  expected  that  every  two  tosses  of  a  coin  will  result  in  one 
head  and  one  tail,  since  other  arrangements  are  possible,  and 
the  laws  of  chance  are  founded  on  the  idea  that  every  possible 
event  will  occur  its  proportionate  number  of  times.  Thus  in 
the  case  of  a  coin  we  have  for  all  possible  events  in  two  tosses, 

Probability  of  2  heads  =  i 

1  head  and  1  tail  =  \ 
"             1  tail  and  1  head  =  i 

2  tails  =  i 

Some  one  of  these  events  must  happen,  so  that  the  total  prob- 
ability is  i+J+l+i,  which  equals  unity,  as  it  should  in  a 
case  of  certainty.  The  probability  of  two  tosses  including  a 
head  and  a  tail  (which  may  occur  in  two  ways)  is  i+}=?,  so 
that  the  proposed  event  is  not  one  that  occurs  at  every  trial, 
as  is  often  inferred. 

An  event  whose  probability  is  extremely  high  will  not  neces- 
sarily happen  on  a  given  occasion,  and  this  failure  to  happen 


DEFINITIONS  AND  PRINCIPLES  251 

does  not  imply  an  error  iifc  the  theory  of  probabilities.  The 
very  fact  that  the  given  probability  is  not  quite  unity  indicates 
the  chance  of  occasional  failures.  Similarly  an  event  with  a 
very  small  probability  will  sometimes  happen,  otherwise  its 
probability  should  be  precisely  and  not  approximately  zero. 

The  probability  of  a  future  event  is  not  affected  by  the 
result  of  events  which  have  already  taken  place.  Thus  if  a  tossed 
coin  has  resulted  in  heads  ten  times  in  succession  it  is  natural 
to  look  on  a  new  toss  as  much  more  likely  to  result  in  tails  than 
in  heads;  but  mature  thought  will  show  that  the  probabilities  are 
still  one-half  and  one-half  for  any  new  toss  that  may  be  made.  The 
confusion  in  such  a  case  comes  from  regarding  the  ten  successive 
heads  as  an  abnormal  occurrence,  whereas,  being  one  of  the 
possible  occurrences,  it  should  happen  in  due  course  along  with 
all  other  possible  events.  If  tails  were  more  likely  to  come  up 
than  heads  in  any  particular  toss,  it  would  imply  some  difference 
of  conditions  instead  of  any  overlapping  influence.  If  the  toss 
of  a  coin  is  ever  regarded  as  a  matter  of  chance,  it  must  always 
be  so  regarded. 


CHAPTER  X 

THE   THEORY   OF   ERRORS 

143.  The  Laws  of  Accidental  Error.  The  mathematical 
theory  of  errors  relates  entirely  to  those  errors  which  are  purely 
accidental,  and  which  therefore  follow  the  laws  of  probability. 
Mistakes  or  blunders,  which  follow  no  law,  and  systematic 
errors,  which  follow  special  laws  for  each  individual  case,  can 
not  be  included  in  such  a  discussion.  If  a  sufficient  number  of 
observations  are  taken  it  is  found  by  experience  that  the  accidental 
errors  which  occur  in  the  results  are  governed  by  the  four  fol- 
lowing laws : 

1.  Plus  and  minus  errors  of  the  same  magnitude  occur  with 
equal  frequency. 

This  law  is  a  necessary  consequence  of  the  accidental  char- 
acter of  the  errors.  An  excess  of  plus  or  minus  errors  would 
indicate  some  cause  favoring  that  condition,  whereas  only  acci- 
dental errors  are  under  consideration. 

2.  Errors  of  increasing  magnitude  occur  with  decreasing  frequency. 
This  law  is  the  result  of  experience,  but  for  mathematical 

purposes  it  is  replaced  by  the  equivalent  statement  that  errors 
of  increasing  magnitude  occur  with  decreasing  facility.  For 
reasons  yet  to  appear  (Art.  146)  the  facility  of  an  error  is  rated 
in  units  that  make  it  proportional  to  the  relative  frequency  with 
which  that  error  occurs  instead  of  equal  thereto. 

3.  Very  large  errors  do  not  occur  at  all. 

This  law  is  also  the  result  of  experience,  but  it  is  not  in 
suitable  form  for  mathematical  expression.  It  is  satisfactorily 
replaced  by  the  assumption  that  very  large  errors  occur  with 
great  infrequency. 

4.  Accidental   errors   are   systematically    modified   by   the   cir- 
cumstances of  observation. 

This  law  is  a  necessary  consequence  of  the  first  three  laws, 
and  emphasizes  the  fact  that  these  three  laws  always  hold  good 

252 


THE  THEORY  OF  ERRORS 


253 


however  much  the  absolute  Values  of  the  errors  may  be  modified 
by  favorable  or  unfavorable  conditions.  The  chief  circumstances 
affecting  a  set  of  observations  are  the  atmospheric  conditions, 
the  skill  of  the  observer,  and  the  precision  of  the  instruments. 

144.  Graphical  Representation  of  the  Laws  of  Error.  The 
four  laws  of  error  are  graphically  represented  in  Fig.  63,  in  which 
the  solid  curve  corresponds  to  a  series  of  observations  taken 
under  a  certain  set  of  conditions,  and  the  dotted  curve  to  a 
senes  of  observations  taken  under  more  favorable  conditions. 
For  reasons  which  will  appear  in  due  course  any  such  curve  is 
called  a  probability  curve.  The  line  XX,  or  axis  of  x,  is  taken 
as  the  axis  of  errors,  and  the  line  AY,  or  axis  of  y,  as  the  axis 
of  facility,  the  point  A  being  taken  as  the  origin  of  coordinates. 
Thus  in  the  case  of  the  solid  curve,  if  the  line  Aa  represents  any 


FIG.  63.— Probability  Curves. 

proposed  error,  then  the  ordinate  ab  represents  the  facility  with 
which  that  error  occurs  in  the  case  assumed.  The  first  law  is 
illustrated  by  making  the  curves  symmetrical  with  reference  to 
the  axis  of  y,  so  that  the  ordinates  are  equal  for  corresponding 
plus  and  minus  values  of  x.  The  second  law  is  illustrated  by  the 
decreasing  ordinates  as  the  plus  and  minus  abscissas  are  increased 
in  length.  The  third  law  does  not  admit  of  exact  representation, 
since  a  mathematical  curve  can  not  have  all  its  ordinates  equal 
to  zero  after  passing  a  certain  point;  a  satisfactory  result  is 
reached,  however,  by  making  all  ordinates  after  a  certain  point 
extremely  small,  with  the  axis  of  x  as  an  asymptote  to  the  curves. 
The  fourth  law  is  illustrated  by  means  of  the  solid  curve  and  the 
dotted  curve,  both  of  which  are  consistent  with  the  first  three 
laws,  but  which  have  different  ordinates  for  the  same  proposed 
error.  Thus  small  errors,  such  as  Aa,  occur  with  greater  frequency 
(or  greater  facility)  in  the  case  of  the  dotted  curve  than  in  the 


254  GEODETIC  SURVEYING 

case  of  the  solid  curve,  as  shown  by  the  ordinate  ac  being  longer 
than  the  ordinate  ab;  while  large  errors,  such  as  Ad,  occur  with 
less  frequency  (or  less  facility)  in  the  case  of  the  dotted  curve 
than  in  the  case  of  the  solid  curve,  as  shown  by  the  ordinate 
de  being  shorter  than  the  ordinate  df. 

145.  The  Two  Types  of  Error.  The  recorded  readings  in 
any  series  of  observations  are  subject  to  two  distinct  types  of 
error.  The  first  type  of  error  includes  all  those  errors  involved 
in  the  making  of  the  measurement,  such  as  those  due  to  imper- 
fect instrumental  adjustments, unfavorable  atmospheric  conditions, 
imperfect  bisection  of  targets,  imperfect  estimation  of  scale 
readings,  etc.  The  second  type  of  error  is  that  involved  in 
the  reading  or  recording  of  the  result,  which  must  be  done  in 
terms  of  some  definite  least  count  which  excludes  all  inter- 
mediate values. 

A  given  reading,  therefore,  does  not  indicate  that  precisely  that 
value  has  been  reached  in  the  process  of  measurement,  but  only 
such  a  value  as  must  be  represented  by  that  reading;  so  that 
a  given  reading  may  be  due  to  any  one  of  an  infinite  number 
of  possible  values  lying  within  the  limits  of  the  least  count. 
Similarly,  the  error  in  the  recorded  reading  does  not  indicate 
that  precisely  that  error  has  been  made  in  the  process  of  measure- 
ment, but  only  such  an  error  as  must  be  represented  by  that 
value;  so  that  the  error  of  the  recorded  reading  may  in  fact 
be  due  to  any  one  of  an  infinite  number  of  possible  errors  lying 
within  the  limits  of  the  least  count.  The  first  type  of  error 
is  the  true  type  or  that  which  corresponds  to  the  accidental 
conditions  under  which  a  series  of  observations  are  made,  while 
the  second  type  is  a  false  type  or  definite  condition  or  limitation 
under  which  the  work  must  be  done.  Thus  in  sighting  at  a  target 
a  number  of  times  the  angular  errors  of  bisection  may  vary 
among  themselves  by  amounts  which  can  only  be  expressed  in 
indefinitely  small  decimals  of  a  second.  If  the  least  count 
recognized  in  recording  the  scale  readings  is  one  second,  however, 
the  recorded  readings  and  the  corresponding  errors  will  vary  among 
themselves  by  amounts  which  differ  by  even  seconds.  The 
probability  curve  of  the  preceding  article  is  based  on  the  first 
type  of  error  only,  and  is  therefore  a  mathematically  con- 
tinuous curve,  since  all  values  of  the  error  are  possible  with  this 
type.  In  speaking  of  the  errors  of  observations,  however,  the 


THE  THEORY  OF  ERRORS  255 

> 

errors  of  the  recorded  values  ^re  in  general  understood,  and  these 
must  necessarily  differ  among  themselves  by  exactly  the  value 
of  the  least  count. 

146.  The  Facility  of  Error.  If  an  instrument  is  correctly 
read  to  any  given  least  count,  no  reading  can  be  in  error  by  more 
than  plus  or  minus  a  half  of  this  least  count;  or,  in  other  words, 
each  reading  is  the  central  value  of  an  infinite  number  of 
possible  values  lying  within  the  limits  of  the  least  count.  If 
a  great  many  observations  are  taken  on  a  given  magnitude,  each 
particular  reading  will  be  found  to  repeat  itself  with  more  or 
less  frequency,  since  all  values  lying  within  a  half  of  the  least 
count  of  that  particular  reading  must  be  recorded  with  the 
value  of  that  reading.  If  the  same  instrument,  however,  carried 
finer  graduations,  with  the  least  count  half  the  previous  value, 
each  reading  would  represent  only  those  values  within  half 
the  previous  limits.  There  would  then  be  twice,  as  many  repre- 
sentative readings,  with  each  one  standing  for  half  as  many 
actual  values  as  with  the  coarser  graduations.  It  is  thus  seen 
that  the  relative  frequency  with  which  a  given  reading  (and 
the  corresponding  error)  occurs,  is  directly  proportional  to  the 
least  count  of  the  instrument,  or  least  count  used  in  recording 
the  readings.  Just  as  each  reading  is  taken  to  represent  an 
infinite  number  of  possible  values  within  the  limits  of  the  least 
count,  so  that  reading  must  correspond  to  an  infinite  number  of 
possible  errors  within  the  same  limits,  each  possible  error  having 
a  different  facility  of  occurrence.  Since  in  the  long  run,  however, 
each  reading  will  be  practically  the  average  of  all  the  values 
that  it  represents,  so  the  facility  of  the  error  due  to  that  reading 
may  be  taken  practically  as  the  average  facility  of  all  the  corre- 
sponding errors.  By  definition  (Art.  143)  the  facility  of  a  given 
accidental  error  is  proportional  to  the  frequency  of  its  occurrence. 
It  is  thus  seen  that  the  relative  frequency  with  which  a  given 
error  (representing  all  possible  errors  due  to  a  given  reading) 
occurs,  is  proportional  to  the  facility  of  that  error.  Since  the 
relative  frequency  with  which  a  given  error  occurs  is  proportional 
to  both  its  facility  and  the  least  count,  it  is  proportional  to 
tyieir  product,  and  is  always  made  equal  to  this  product  by  using 
a  suitable  scale  of  facility.  The  facility  of  a  given  error  is  hence 
equal  to  the  relative  frequency  of  occurrence  of  that  error  divided 
by  the  least  count. 


256 


GEODETIC  SURVEYING 


147.  The  Probability  of  Error.  By  the  probability  of  an 
error  is  meant  the  relative  frequency  of  its  occurrence.  Thus 
in  the  measurement  of  an  angle,  if  a  given  error  occurred  (on 
the  average)  27  times  in  1000  observations,  then  the  probability 
that  an  additional  measurement  would  be  in  error  by  that  same 
amount  would  be  T?^hr-  The  probability  of  a  given  error  being 
identical  with  its  relative  frequency  of  occurrence  is  hence  (Art. 
146)  equal  to  the  product  of  the  facility  of  that  error  by  the  least 
count.  The  probability  of  error  for  a  certain  set  of  conditions 
is  illustrated  in  Fig.  64.  In  this  figure  the  spaces  da,  ae,  eb,  and 
bf  are  each,  equal  to  one-half  of  the  least  count.  The  probability 
that  an  error  A  a  will  occur  is  hence,  in  accordance  with  the 
above  principles,  equal  to  the  product  of  am  (the  facility)  by 


c  A     d    a    e     b   f         u  X 

FIG.  64.—  The  Probability  of  Error. 

de  (the  least  count).     As  the  least  count  is    always  very  small, 
we  may  write  without  appreciable  error, 


Probability  of  error  Aa  = 


dste. 


But  (Art.  145)  the  error  Aa  in  the  recorded  reading  includes  all 
the  possible  errors  lying  between  Ad  and  Ae,  that  is,  within 
half  the  least  count  each  way  from  A  a.  The  area  dste  therefore 
represents  the  probability  that  the  actual  error  committed  lies 
between  the  values  Ad  and  Ae.  Similarly  the  area  etuf  represents 
the  probability  of  an  actual  error  between  the  values  Ae  and 
Af.  The  probability  that  an  actual  error  shall  lie  either  between 
Ad  and  Ae  or  between  Ae  and  Af  (compound  event,  Art.  140), 
or  in  other  words  between  Ad  and  Af,  is  equal  to  the  sum  of  the 
two  separate  probabilities,  that  is,  to  the  combined  area  dsuf. 
Or,  in  general,  the  probability  that  an  error  shall  fall  between 
any  two  values  Ac  and  Ag,  is  represented  by  the  area  included 
between  the  corresponding  ordinates  cr  and  gv.  On  account 


THE  THEORY  OF  ERRORS  257 

of  this  characteristic  property  the  curve  of  facilities  is  commonly 
called  the  probability  curve.  Strictly  speaking  the  ordinates 
limiting  the  area  can  only  occur  at  certain  equally  spaced  intervals 
depending  on  the  least  count,  but  no  material  error  is  ever  intro- 
duced by  drawing  them  at  any  points  whatever. 

148.  The  Law  of  the  Facility  of  Error  is  that  law  which  con- 
nects all  the  possible  errors  in  any  set  of  observations  with  their 
corresponding  facilities,    and   is   expressed   analytically   by  the 
equation  of  the  probability  curve.     The  law  which  governs  the 
occurrence   of   errors  in  any  particular  set   of  observations  is 
necessarily  unknown  and  beyond  determination,  being  the  com- 
bined result  of  an  uncertain  number  of  variable  and  unknown 
causes.     Fortunately,  however,  it  is  found  by  experience  that 
there  is  one  particular  form  of  law  which  (with  proper  constants) 
very  closely  represents  the  facility  of  error  in  all  classes  of  obser- 
vations.    This  form  of  law  is  that  which  is  in  accordance  with 
the  assumption  that  the  arithmetic  mean  of  the  observed  values 
is  the  most  probable  value  when  the  same  magnitude  has  been 
observed  a  large  number  of  times  under  the  same  conditions. 
The  same  form  of  law  being  accepted  as  satisfactory  in  all  cases, 
therefore,  the  law  for  any  particular  case  is  determined  by  the 
substitution  of  the  proper  constants. 

149.  Form  of  the  Probability  Equation.     If  x  represents  any 
possible  error  and  y  the  facility  of  its  occurrence,  we  may  write 

y  =<£(*)>  .    ;.;".'    .  >    •    .    .    (6) 

which  is  read  y  equals  a  function  of  x.  When  the  form  of  this 
function  has  been  determined  the  expression  will  be  the  general 
equation  of  the  probability  curve.  Since  the  probability  that 
the  error  x  (of  a  recorded  reading)  will  occur  is  equal  (Art.  147) 
to  its  facility  multiplied  by  the  least  count,  we  have 

P  =  yAx  =  4>(x)Jx,     .     .'    .....     (7) 

in  which  P  is  the  probability  of  the  occurrence  of  the  error  x, 
and  ^x  is  the  least  count.  If  x\,  X2,  .  .  .  xn  are  the  true  errors  in 
the  observed  values  of  any  magnitude  Z,  and  Pi,  P2,  .  .  .  Pn 
are  the  corresponding  probabilities  of  occurrence,  we  thus  have 

PI  =  </>(xi)Jx,     P2  =  <t>(x2)Jx,     etc. 


258  GEODETIC  SURVEYING 

The  probability  P  of  the  occurrence  of  this  particular  series 
of  errors,  x\,  x2,  etc.,  in  a  set  of  observations  of  equal  weight, 
being  a  concurrent  event  (Art.  141),  is  equal  to  the  product 
of  the  individual  probabilities,  giving 


.     .     .     .     (8) 
whence 


log  P  =  log  0(zi)  +  log  </>(x2).  .  .  +  log  <f>(xn)  +  n  log  Ax.    (9) 

The  true  value  of  the  unknown  quantity  Z,  and  the  errors 
Xi,  x2,  etc.,  can  never  be  known.  Any  assumed  value  of  Z  will 
result  in  a  particular  series  of  values  v\,  v2,  etc.,  for  the  errors 
of  the  several  observations.  That  value  of  Z  will  be  the  most 
probable  which  produces  the  series  of  errors  which  has  the 
highest  probability  of  occurrence.  Replacing  the  true  errors 
Xi,  x2)  etc.,  in  Eq.  (9)  by  the  variable  errors  v\,  v2,  etc.,  and 
making  the  first  differential  coefficient  equal  to  zero  to  obtain 
a  maximum  value  of  P}  we  have 


d  log  ,  n 

I  7  •     .     .      "|  7  -     \J  %  \  JL\J  I 

i  dv2  dvn 

which  may  be  written 

=  p. 


But  it  has  already  been  decided  (Art.  134)  that  the  arithmetic 
mean  of  such  a  series  of  observed  values  is  the  most  probable 
value  of  the  quantity  observed.  The  adoption  of  the  arith- 
metic mean  as  the  most  probable  value,  however,  requires 
the  algebraic  sum  of  the  residuals  (Art.  135)  to  reduce  to  zero; 
whence 

VL  +  V2  .  .  .  +  vn  =  0  .......     (12) 

Since  v\,  v2,  etc.,  are  the  result  of  chance,  and  hence  independent 
of  each  other,  it  follows  from  Eq.  (12)  that  the  coefficients  of 
»i,  etc.,  in  Eq.  (11)  must  all  have  the  same  value.  Representing 
this  unknown  value  for  any  particular  set  of  observations  by  the 


THE  THEORY  OF  EREORS  259 

constant  k,  we  have  as  the  general  condition  which  makes  the 
arithmetic  mean  the  most  probable  value, 


vdv 
whence  by  transposition 

d  log  <j>(v)  =  kvdv. 
Integrating  this  equation 

log  0(«>)  =  \kv2  +  log  c, 


in  which  log  c  represents  the  unknown  constant  of  integration. 
Passing  to  numbers,  we  have 

<K»)  =«***,      .     .    .    .   ...     ,    (13) 

in  which  e  equals  the  base  of  the  Naperian  system  of  logarithms. 
It  is  necessary  at  this  point  to  remember  that  the  probability 
of  the  occurrence  of  a  given  error  does  not  involve  the  question 
as  to  whether  we  are  right  or  wrong  in  assuming  that  an  error  of 
that  value  has  occurred  in  a  particular  observation.  Thus  in 
the  preceding  discussion  the  probabilities  assigned  to  the  assumed 
values  of  PI,  t>2,  etc.,  are  the  probabilities  for  true  errors  of  these 
values,  regardless  of  whether  such  errors  have  or  have  not  occurred 
in  the  given  case.  It  is  of  the  utmost  importance,  therefore,  to 
realize  that  Eq.  (13)  is  not  based  on  the  assumption  that  the 
error  v  has  occurred,  but  is  a  general  statement  of  fact  concern- 
ing any  true  error  whose  magnitude  is  v.  Replacing  v  in  Eq.  (13) 
by  x,  the  adopted  symbol  for  true  errors,  we  have 


but  from  equation  (6) 

y  = 

whence 

y    = 


260  GEODETIC  SURVEYING 

Since  the  facility  y  decreases  as  the  numerical  value  of  x  increases, 
it  follows  that  i  k  is  essentially  negative,  and  it  is  therefore 
commonly  replaced  by  —  h2.  Making  this  substitution,  we  have 


(14) 


in  which  y  equals  the  facility  with  which  any  error  x  occurs, 
c  and  h  are  unknown  constants  depending  on  the  circumstances 
of  observation,  and  e  is  the  base  of  the  Naperian  system  of  log- 
arithms. Though  correct  in  apparent  form,  Eq.  (14)  must  not 
yet  be  regarded  as  the  general  equation  of  the  probability  curve, 
since  the  quantities  c  and  h  appear  as  arbitrary  constants, 
whereas  t  wi  1  be  shown  in  the  next  article  that  these  values  are 
dependent  on  each  other. 

150.  General  Equation  of  the  Probability  Curve.  The  proba- 
bility that  an  error  shall  fall  between  any  two  given  values 
(Art.  147)  is  equal  to  the  area  between  the  corresponding  ordi- 
nates  of  the  probability  curve.  The  probability  that  an  error  shall 
fall  between  —  GO  and  +  oo  is  therefore  equal  to  the  entire  area 
of  the  curve.  But  it  is  absolutely  certain  that  any  error  which 
may  occur  will  fall  between  these  extreme  limits,  and  the  proba- 
bility of  a  certain  event  (Art.  138)  is  equal  to  unity.-  The  entire 
area  of  any  curve  represented  by  Eq.  (14)  must  therefore  be  equal 
to  unity.  Since  all  probability  curves  have  the  same  total  area, 
it  follows  that  any  change  in  h  will  require  a  compensating  change 
in  c;  or,  in  other  words,  c  must  be  a  function  of  h.  The  general 
expression  for  the  area  of  any  plane  curve  is 


= J  ydx. 


Substituting  the  value  of  y  from  Eq.  (14) 

A  == 


The  probability  P  that  an  error  x  will  fall  between  the  limits  a 
and  6,  is  therefore 


p  =  CbCe-h2*zdx,      ...          .     (15) 

./ n 


THE  THEORY  OF  ERRORS  261 

and  between  the  limits  —  oo  qgad  +  oo  ,  is 

P  =  f  °°  ce-^dx  =  c  f  °°  e~h'*'dx. 

J-<x>  J  -oo 

But  this  probability,  being  a  certainty,  equals  unity;  whence 

l=c 
or 


I/*  00 
/ 

c  =J_  f  Xdx' 


The  second  member  of  this  equation  is  a  definite  integral  whose 
evaluation  by  the  methods  of  the  calculus  (for  which  such  works 
should  be  seen)  gives 

v* 


°°  e-;-vdx  = 

—  00  fl 


hence 

!_ 
c 

and 


which  substituted  in  Eq.  (15)  gives  for  the  probability  P  that  an 
error  x  will  fall  between  any  limits  a  and  6, 

P  =-7=CJe-'^dx.    v   .  (16) 


Also  substituting  the  above  value  of  c  in  Eq.  (14)  we  have  for  the 
general  equation  of  the  probability  curve 


in  which  y  is  the  facility  with  which  any  error  x  occurs,  e 
(  =  2.7182818)  is  the  base  of  the  Naperian  system  of  logarithms, 
and  h  (called  the  precision  factor)  is  a  constant  depending  on  the 
circumstances  of  observation.  The  constant  h  is  the  only  element 


262  GEODETIC  SURVEYING 

in  Eq.  (17)  which  can  vary  with  the  precision  of  the  work,  and 
therefore  of  necessity  becomes  the  measure  of  that  precision. 
151.  The  Value  of  the  Precision  Factor.  The  general  equa- 
tion of  the  probability  curve  is  given  by  Eq.  (17),  but  the  definite 
equation  for  any  particular  set  of  observations  is  not  known 
until  the  corresponding  value  of  h  has  been  determined.  The 
probability  that  an  error  x  will  occur  (Art.  149)  is 


P    =  yjx    =  (j)(x)Jx. 

Substituting  the  value  of  y  from  Eq.  (17), 


P  =  -=  e~h^Ax  =  (j>(x)Ax.  .  (18) 

VTT 

With  an  infinite  number  of  observations  any  residual  vl  would  be 
infinitely  close  to  the  corresponding  true  error  x\,  and  the  relative 
frequency  with  which  v±  occurred  would  not  differ  appreciably 
from  PI.  The  value  of  h  for  any  particular  case  could  thus  be 
found  from  Eq.  (18)  by  substituting  these  values  for  P  and  x. 
As  the  number  of  observations  is  always  limited,  however,  the 
best  that  can  be  done  is  to  find  the  most  probable  value  of  h 
for  the  given  case.  The  probability  that  a  given  set  of  errors 
has  occurred  is,  by  Eq.  (8), 


P 

But  from  Eqs.  (6)  and  (17) 


so  that 

P  —  /  '"   \  p~i 

f  7=  I 

and 

log  P  =  n  log  h  -  h*2x2  +  n  log  Ax  - 1-  log  TT; 

whence  by  making  the  first  derivative  with  respect  to  h  equal  to 
zero 

n 


=  0. 
n 


THE  THEORY  OF  ERRORS  263 

i 

Solving  for  h  we  have 

•  •  •  •  •  •  •  (19) 


in  which  n  is  the  number  of  observations  taken,  and  2z2  is  the 
sum  of  the  squares  of  the  true  errors  which  have  occurred.  The 
true  errors,  however,  can  never  be  known,  and  formula  (19)  must 
therefore  be  modified  so  as  to  give  the  most  probable  value  of  h 
that  can  be  determined  from  the  residual  errors.  A  discussion 
of  this  condition  is  beyond  the  scope  of  this  book,  but  for  observa- 
tions of  equal  (or  unit)  weight  results  in  the  formula 


(20) 


in  which  n  as  before  is  the  number  of  observations  that  have  been 
taken,  and  ^v2  is  the  sum  of  the  squares  of  the  residual  errors. 
For   observations  of   unequal  weight  (Art.  133)  formula  (19) 
becomes 


h  =  Jfrv-1,, (21) 


in  which  Spy2  is  the  sum  of  the  weighted  squares  of  the  residuals, 
and  h  as  before  is  the  precision  factor  for  observations  of  unit 
weight. 

For  the  general  case  of  indirect  observations  (Art.  158)  on  inde- 
pendent quantities,  that  is,  with  no  conditional  equations  (Art.  131), 
formula  (19)  becomes 


(22) 


in  which  n  is  the  number  of  observation  equations,  q  is  the  number 
of  unknown  quantities,  ^pv2  is  the  sum  of  the  weighted  squares 
of  the  residuals,  and  h  is  the  precision  factor  for  observations 
of  unit  weight. 

For  the  general  case  of  indirect  observations  involving  con- 
ditional equations,  formula  (19)  becomes 


h  = 


264 


GEODETIC  SURVEYING 


in  which  c  is  the  number  of  conditional  equations,  n  is  the  number 
of  observation  equations,  q  is  the  number  of  unknown  quantities, 
Zpv2  is  the  sum  of  the  weighted  squares  of  the  residuals,  and  h  is 
the  precision  factor  for  observations  of  unit  weight.  As  will  be 
understood  later  (Art.  166),  the  number  of  independent  unknowns 
is  always  reduced  by  an  amount  which  equals  the  number  of 
conditional  equations,  so  that  q  in  Eq.  (22)  is  simply  replaced  by 
(q  -  c)  in  Eq.  (23). 

152.  Comparison  of  Theory  and  Experience.  In  the  Funda- 
menta  Astronomios,  Bessel  gives  the  following  comparison  of  theory 
and  experience.  In  a  series  of  470  observations  by  Bradley  on 
the  right  ascensions  of  Sirius  and  Altair  the  value  of  h  was  found 
to  be  1.80865,  giving  rise  to  the  following  table: 


Numerical  Errors  between 

Probability  of 
Errors. 

Number  of  Errors 

By  Theory. 

By  Experience. 

// 
0.0 

0.1 

0.2018 

94.8 

94 

0.1 

0.2 

0.1889 

88.8 

88 

0.2 

0.3 

0.1666 

78.3 

78 

0.3 

0.4 

0.1364 

64.1 

58 

0.4 

0.5 

0.1053 

49.5 

51 

0.5 

0.6 

0.0761 

35.8 

36 

0.6 

0.7 

0.0514 

24.2 

26 

0.7 

0.8 

0.0328 

15.4 

14 

0.8 

0.9 

0.0194 

9.1 

10 

0.9 

1.0 

0.0107 

5.0 

7 

1.0 

00 

0.0106 

5.0 

8 

Totals  

1.0000 

470.0 

470 

The  last  column  in  this  table  tacitly  assumes  that  the  true  errors 
do  not  differ  materially  from  the  residual  errors,  the  true  errors 
being  of  course  unknown.  The  agreement  of  theory  and  expe- 
rience is  very  satisfactory. 

There  are  two  important  points  to  be  observed  in  applying 
the  theory  of  errors  to  the  results  obtained  in  practical  work. 


THE  THEORY  OF  ERRORS  265 

i 

In  the  first  place,  the  theory  of-errors  presupposes  that  a  very  large 
number  of  observations  have  been  made.  It  is  customary,  how- 
ever, to  apply  the  theory  to  any  number  of  observations,  however 
limited.  It  is  evident  in  such  cases  that  reasonable  judgment 
must  be  used  in  interpreting  the  results  obtained  by  the  applica- 
tion of  the  theory.  In  the  second  place,  the  theory  of  errors  is 
the  theory  of  accidental  errors.  It  is  in  general  impossible  to 
entirely  prevent  systematic  errors  in  a  process  of  observation; 
and  such  errors  can  not  be  discovered  or  eliminated  by  any  num- 
ber of  observations,  however  great,  if  the  circumstances  of  observa- 
tion remain  unchanged.  The  theory  of  errors,  therefore,  makes 
no  pretense  of  discovering  the  truth  in  any  case,  but  only  to 
determine  the  best  conclusions  that  can  be  drawn  from  the  observa- 
tions that  have  been  made. 


CHAPTER  XI 

MOST   PROBABLE  VALUES   OF  INDEPENDENT  QUANTITIES 

153.  General  Considerations.     In   accordance  with  the  dis- 
cussions of  the  previous  chapter  it  is  evident  that  the  true  value 
of  an  observed  quantity  can  never  be  found.     Adopting  any 
particular  value  for  the  observed  quantity  is  equivalent  to  assum- 
ing that  a  certain  series  of  errors  has  occurred  in  the  observed 
values.     Manifestly  the  most  probable  value  of  the  observed 
quantity  is  that  which  corresponds  to  the  most  probable  series 
of  errors;  or,  in  other  words,  that  series  of  errors  which  has  the 
highest  probability  of  occurrence.     It  is  therefore  by  means  of 
the  theory  of  errors  (Chapter  X)  that  rules  are  established  for 
determining  the  most  probable  values  of  observed   quantities. 

154.  Fundamental  Principle  of  Least  Squares.    For  the  general 
equation  of  the  probability  curve,  Eq.  (17),  Art.  150,  we  have 


in  which  y  is  the  facility  of  occurrence  of  any  error  x  under  the 
conditions  represented  by  the  precision  factor  h.  The  probability 
that  any  error  x  will  occur  (Art.  147)  is  equal  to  its  facility  multi- 
plied by  the  least  count,  or 

P  =  yJx. 

Hence  if  x\,  #2,  •  •  •  £n  are  the  errors  in  the  observed  values 
of  any  magnitude  Z,  and  PI,  Po,  .  .  .  Pn  are  the  corre- 
sponding probabilities  of  occurrence,  we  have 

2/2  =  -=  e~h*x**,    etc., 


- 
Vjc  VTT 

and 

etc. 

266 


PROBABLE  VALUES  OF  INDEPENDENT  QUANTITIES     267 

The  probability  P  of  the  occurrence  of  this  particular  series  of 
errors  x\,  x2,  etc.,  in  the  given  set  of  observations,  being  a  con- 
current event  (Art.  141),  is  equal  to  the  product  of  the  individual 
probabilities,  giving 

P  = 


This  equation  is  true  for  any  proposed  series  of  errors,  and 
hence  for  that  series  of  residual  errors  vit  v2,  .  .  .  vn,  which 
results  from  assigning  the  most  probable  value  to  the  observed 
quantity.  In  this  case  Hz2  becomes  5>2,  and  we  have 

(24) 


But  (Art.  153)  the  most  probable  value  of  the  observed  quantity 
corresponds  to  that  series  of  errors  which  has  the  highest  prob- 
ability of  occurrence.  The  most  probable  value  z  of  any  observed 
quantity  Z,  therefore,  requires  P  in  Eq.  (24)  to  be  a  maximum, 
and  this  in  turn  requires  2>2  to  be  a  minimum.  We  thus  have  the 
following 

PRINCIPLE:  In  observations  of  equal  precision  the  most  probable 
values  of  the  observed  quantities  are  those  that  render  the  sum  of  the 
squares  of  the  residual  errors  a  minimum. 

It  is  on  account  of  this  principle  that  the  Method  of  Least 
Squares  has  been  so  named. 

155.  Direct  Observations  of  Equal  Weight.  A  direct  observa- 
tion (Art.  133)  is  one  that  is  made  directly  on  the  quantity  whose 
value  is  to  be  determined.  When  the  given  magnitude  is  measured 
a  number  of  times  under  the  same  conditions  (as  represented 
by  the  same  precision  factor  h  in  the  probability  curve) ,  the  results 
obtained  are  said  to  be  of  equal  weight  or  precision.  In  such  a  case 
the  most  probable  value  of  the  quantity  sought  mu^t  accord  with 
the  principle  of  the  previous  article,  that  is,  the  sum  of  the  squares 
of  the  residual  errors  must  be  a  minimum. 

Let  z  =  the  most  probable  value  of  a  given  magnitude; 
n  =  the  number  of  measurements  taken; 
MI,  M2,  •  .  .  Mn  =  the  several  measured  values; 

then  (Art.  154) 

(Mi  -  z,2  +  (M2  -  z]2  .  .  .  +  (MH  -  z)2  =  a  minimum. 


268  GEODETIC  SURVEYING 

Placing  the  first  derivative  equal  to  zero, 

2(M,  -  z)  +  2  (M2  -  z)  .  .  .  +  2(Mn  -  2)  =0; 
whence 

(Af  !   +  M2  .  .  .  +  M  n)    ~  ttZ   =   0, 

and 


_  Ml  +  M2  .  .  .  +  Mn  _ 

—  "~;      '    '    '    (25) 


or,  expressed  in  words,  in  the  case  of  direct  observations  of  equal 
weight  the  most  probable  value  of  the  unknown  quantity  is  equal 
to  the  arithmetic  mean  of  the  observed  values.  The  above 
discussion,  however,  must  not  be  regarded  as  a  proof  of  this 
principle  of  the  arithmetic  mean,  since  (Art.  149)  this  very  prin- 
ciple was  one  of  the  conditions  under'  which  the  equation  of 
the  probability  curve  was  deduced.  Eq.  (25)  therefore  simply 
shows  that  the  equation  of  the  probability  curve  is  correct  in  form 
and  consistent  with  this  principle. 

Example.  The  observed  values  (of  equal  weight)  of  an  angle  A  are 
29°  21'  59".l,  29°  22'  06".4,  and  29°  21'  58".l.  What  is  the  most  probable 
value? 

29°  21'  59".l 
29  22  06  .4 
29  21  58  .1 


3)88    06    03  .6 
29    22    01  .2 

The  most  probable  value  is  therefore  29°  22'  01  ".2. 

156.  General  Principle  of  Least  Squares.  When'  a  given 
magnitude  is  measured  a  number  of  times  under  different  con- 
ditions (so  that  the  precision  factor  corresponding  to  some  of  the 
observations  is  not  the  same  for  all  of  them) ,  the  results  obtained 
are  said  to  be  of  unequal  weight  or  precision.  In  accordance  with 
the  sense  in  which  weights  are  understood  (Art.  133),  an  observa- 
tion assigned  a  weight  of  two  means  it  is  considered  as  good  a 
determination  as  the  arithmetic  mean  of  two  observations  of 
unit  weight,  and  so  on.  It  is  immaterial  whether  any  one  of  the 
observed  values  is  considered  of  unit  weight,  as  this  is  merely  a 
basis  of  comparison. 


PROBABLE  VALUES  OF  INDEPENDENT  QUANTITIES    269 

Let  2  =  the  moet  probable  value  of  a  given  magni- 

tude; 

MI,  M2,  etc.  =  the  values  of  the  several  measurements; 
Pi,  p2,  etc.  =  the  respective  weights  of  these  measure- 
ments; 
api,  ap2,  etc.  =  the  corresponding  integral  weights  due  to 

the  use  of  the  factor  a; 

mi',  mi",  etc.  =  the  api  unit  weight  components  of  MI 
when    considered    as   an   arithmetical 
mean; 
m2',  m2",  etc.  =  similarly  for  M2}  and  so  on; 

vij  V2,  etc.  =  the  residuals  due  to  MI,  M2}  etc.; 

then,  as  in  Art.  134,  we  have 
MI  = 

Mo  = 

ap2  ap2 3 

V,™  V^/VT.      71 /f  Vx^TI/f 

....     (28) 

The  value  of  2  thus  obtained  is  evidently  independent  of  any 
particular  set  of  values  that  may  be  assigned  to  the  components 
mi',  m-i",  etc.,  the  components  m2',  m2",  etc.,  and  so  on.  Since 
these  various  components  are  all  of  equal  weight  we  must  have 
in  accordance  with  Art.  154, 

2(2  -  mi)2  +  2(2  -  m2)2  ...  +  2(2-  mn)2  =  a  minimum,  (27) 

as  a  criterion  that  must  be  satisfied  when  2  is  the  most  probable 
value  of  the  quantity  Z.  But,  in  accordance  with  Eq.  (26), 
this  criterion  must  determine  the  same  value  of  z  no  matter  what 
particular  sets  of  values  may  be  substituted  for  the  components 
w&i'j  wii",  etc.,  m2',  m2",  etc.,  and  so  on.  Adopting,  therefore, 
the  particular  sets  of  values 

m\    =  m/'  =  .  .  .  =  MI, 
etc.  etc., 


270  GEODETIC  SURVEYING 

whence 

S(z  -  mi)2  =  api  (z  -  Mi)2  =  api-vf, 

S(z  -  m2)2  =  ap2  (z  -  M2)2  =  ap2-v22, 
etc.  etc., 

and  substituting  in  Eq.  (27),  we  have 

api'Vi2  +  ap2-v22.  .  .  +  apn-vn2  =  a  minimum; 
or,  dividing  out  the  common  factor  a, 

PiVi2  +  P2V22.  .  .  +  pn  vn2  =  a  minimum.     .     .     (28) 
We  thus  have  the  following 

GENERAL  PRINCIPLE:  In  observations  of  unequal  precision  the 
most  probable  values  of  the  observed  quantities  are  those  that  render 
the  sum  of  the  weighted  squares  of  the  residual  errors  a  minimum. 

157.  Direct  Observations  of  Unequal  Weight.  When  a  given 
magnitude  is  directly  measured  a  number  of  times  it  may  be 
necessary  to  assign  different  weights  to  the  results  obtained,  on 
account  of  some  change  in  the  conditions  governing  the  measure- 
ments. In  such  a  case  the  most  probable  value  of  the  quantity 
sought  must  accord  with  the  principle  of  the  previous  article, 
that  is,  the  sum  of  the  weighted  squares  of  the  residual  errors 
must  be  a  minimum. 

Let  z  =  the  most  probable  value  of  a  given  magnitude; 
MI,  M2)  .  .  Mn  =  the  several  measured  values; 
Pi,  p2,  .  .  .  pn   =  the  corresponding  weights; 

then  (Art.  156) 

pi(Mi  —  z)2  +  p2(M2  —  z)2  .  .  .  +  pn(Mn  —  z)2  =  a  minimum. 
Placing  the  first  derivative  equal  to  zero, 

2pi(Mi  -z)  +  2p2(M2  -  z)  .  .  .  +  2pn(Mn  -  z)  =  0; 


PROBABLE  VALUES  OF  INDEPENDENT  QUANTITIES    271 

.         9 
whence  ^ 

(Pi  M!  +  p2M2  .  .  .  +  pnMn)  -  (pi  +  p2  .  .  .  +  Pn)  z  =  0, 
and 


Z  = 


p2M2.  .  .  +pnMn 

-  = 


Pi    +   P2  •    •    •    +   Pn 


or,  expressed  in  words,  in  the  case  of  direct  observations  of  unequal 
weight  the  most  probable  value  of  the  unknown  quantity  is  equal 
to  the  weighted  arithmetic  mean  of  the  observed  values.  The 
above  discussion,  however,  must  not  be  regarded  as  a  proof  of 
this  principle  of  the  weighted  arithmetic  mean,  since  Eq.  (29) 
is  deduced  from  a  principle  based  in  part  on  the  truth  of  Eq.  (26), 
which  is  identical  with  Eq.  (29).  As  the  truth  of  Eq.  (26)  is 
established  in  Art.  156,  however,  Eq.  (29)  shows  that  the  general 
principle  of  least  squares  leads  to  a  correct  result  in  a  case  where 
the  answer  is  already  known. 

Example.  The  observed  values  for  the  length  of  a  certain  base  line  are 
4863.241  ft.  (weight  2),  and  4863.182  ft.  (weight  1).  What  is  the  most 
probable  value? 

4863.241  X  2  =  9726.482 
4863.182  X  1  =  4863.182 


3)14589.664 
4863.221 

The  most  probable  value  is  therefore  4863.221  ft. 

158.  Indirect  Observations.  An  indirect  observation  is  one 
that  is  made  on  some  function  of  one  or  more  quantities,  instead 
of  being  made  directly  on  the  quantities  themselves.  Thus  in 
measuring  an  angle  by  repetition  the  observation  is  indirect,  as 
the  angle  actually  read  is  not  the  angle  sought,  but  some  multiple 
thereof.  Similarly  when  angles  are  measured  in  combination 
the  observations  are  indirect,  since  the  values  of  the  individual 
angles  must  be  deduced  from  the  results  obtained  by  some  pro- 
cess of  computation. 

An  observation  equation  is  an  equation  expressing  the  function 
observed  and  the  value  obtained.  Thus  if  x,  y,  etc.,  represent 
the  unknown  quantities  whose  values  are  to  be  deduced  from  the 


272  GEODETIC  SURVEYING 

observation,  we  may  have  as  observation  equations  such  expres- 
sions as 

6z  =  185°  19'  40", 
or 

7x  +  IQy  -  3z  =  65.73, 

according  to  the  function  observed. 

In  general  the  observation  equations  which  occur  in  geodetic 
work  may  be  written  in  the  following  form : 


(30) 


+  biy  +  ctf  .  .  .  =  MI  (weight  pi)  } 
a2x  +  b2y  +  c2z  .  .  .  =  M 2  (weight  p2 

anx  +  bny  +  cnz  . .  .  =  Mn  (weight  pn)  J 


in  which  ai,  a2,  61,  b2  etc.,  are  known  coefficients;  x,  y,  etc.,  are 
the  unknown  quantities;  Mi,  M2,  etc.,  are  the  observed  values; 
and  pi,  p2,  etc.,  are  the  respective  weights  of  these  values.  If  the 
number  of  observation  equations  is  less  than  the  number  of  unknown 
quantities,  the  values  of  x}  y,  z,  etc.,  can  not  be  found,  nor  even 
their  most  probable  values.  If  the  number  of  observation  equa- 
tions equals  the  number  of  unknown  quantities,  the  equations 
may  be  solved  as  simultaneous  equations,  and  each  equation  will 
be  exactly  satisfied  by  the  values  obtained  for  x,  y,  z,  etc.,  even 
though  these  values  are  not  the  true  values  sought.  If  the  num- 
ber of  observation  equations  exceeds  the  number  of  unknown 
quantities  there  will  in  general  be  no  values  of  x,  y,  z,  etc.,  which 
will  exactly  satisfy  all  the  equations,  on  account  of  the  unavoidable 
errors  of  observation.  Hence  if  the  most  probable  values  of  the 
unknown  quantities  be  substituted  the  equations  will  not  be 
exact1  y  satisfied,  but  will  reduce  to  small  residuals  vi,  V2,  v^,  etc. 
If,  therefore,  x,  y,  z,  etc.,  be  understood  to  mean  the  most  probable 
values  of  these  quantities,  we  will  have 

diX  +  biy  +  CiZ  .  .  .  —  MI  =  Vi  (weight  pi) 

d2x  +  b2y  +  c2z  .  .  .  —  M2  =  v2  (weight  p2) 

(31) 

anx  +  bny  +  cnz  ...  -  Mn  =  vn  (weight  pn) 


PROBABLE  VALUES  OF  INDEPENDENT  QUANTITIES    273 

By  a  consideration  of  these  equations,  together  with  any  special 
conditions  which  must  be  satisfied,  rules  may  be  established  for 
finding  the  most  probable  values  of  the  unknown  quantities  in 
all  cases  of  indirect  observations. 

159.  Indirect  Observations  of  Equal  Weight  on  Independent 
Quantities.  An  independent  quantity  is  one  whose  value  is 
independent  of  the  value  of  any  other  quantity  under  considera- 
tion. Thus  in  a  line  of  levels  the  elevation  of  any  particular 
bench  mark  bears  no  necessary  relation  to  the  elevation  of  any 
other  bench  mark ;  whereas  in  a  triangle  the  three  angles  are  not 
independent  of  each  other,  as  their  sum  must  necessarily  equal 
180°. 

In  the  case  of  indirect  observations  of  equal  weight  on  inde- 
pendent quantities,  the  most  probable  values  of  the  unknown 
quantities  are  found  by  a  direct  application  of  the  method  of 
normal  equations.  A  normal  equation  is  an.  equation  of  condi- 
tion which  determines  the  most  probable  value  of  any  one  unknown 
quantity  corresponding  to  any  particular  set  of  values  assigned  to 
the  remaining  unknowns.  A  normal  equation  must  therefore 
be  specifically  a  normal  equation  in  x,  or  in  y,  etc.  By  forming 
a  normal  equation  for  each  of  the  unknowns  there  will  be  as  many 
equations  as  unknown  quantities.  The  solution  of  these  equa- 
tions as  simultaneous  will  give  a  set  of  values  for  the  unknowns 
in  which  each  value  is  the  most  probable  that  is  consistent  with 
the  remaining  values,  which  can  only  be  the  case  when  all  the 
values  are  simultaneously  the  most  probable  values  of  the  unknown 
quantities. 

To  establish  a  rule  for  forming  the  normal  equations  in  the 
case  of  equal  weights  let  us  re-write  Eqs.  (31),  omitting  the 
weights,  thus: 

aix  +  biy  +  ciz   .  .  .  -  Ml  =  v 


b2y  +  c2z  .  .  .  —  M2  =  t'2 


anx  +  bny  +  cnz  .  .  .  -  Mn  =  v, 


(32) 


In  accordance  with  Art.  154  the  most  probable  values  of  the 
unknown  quantities  are  those  which  give 

t>i2  +  V22  .  .  •  +  vn2  =  a  minimum. 


274  GEODETIC  SURVEYING 

Since  (in  lorming  the  normal  equations)  the  most  probable  value 
of  x  is  desired  for  any  assumed  set  of  values  for  the  remaining 
unknowns,  we  place  the  first  derivative  with  respect  to  x  equal 
to  zero;  whence,  omitting  the  common  factor  2,  we  have 


But  from  Eqs.  (32),  under  the  given  assumption  of  fixed  values 
for  all  quantities  excepting  x,  we  obtain 

dvi  dv2 

~dx   "=  °"    ~dx  "=  a2'  etc'; 

whence  by  substitution, 

a\vi  +  a,2V2  •'••»'+  anVn  =  0  =  normal  equation  in  x. 
In  a  similar  manner  we  have 

+  62^2  ••••+•  bnvn  =  0  =  normal  equation  in  y; 

-+-  C2V2  •  •  •  +  cnvn  =  0  =  normal  equation  in  z\ 
etc.,  etc.; 

and  hence  for  forming  the  several  normal  equations  in  the  case 
of  indirect  observations  of  equal  weight  on  independent  quan- 
tities, we  have  the  following 

RULE  :  To  form  the  normal  equation  for  each  one  of  the  unknown 
quantities,  multiply  each  observation  equation  by  the  algebraic 
coefficient  of  that  unknown  quantity  in  that  equation,  and  add  the 
results. 

Having  formed  the  several  normal  equations,  their  solution 
as  simultaneous  equations  gives  the  most  probable  values  of  the 
unknown  quantities. 

Example  1.     Given  the  observation  equation 
Qx  =  90°  15'  30". 

In  applying  'the  above  rule  to  this  case  we  would  have  to  multiply  the  whole 
equation  by  6,  and  then  divide  by  36  to  obtain  the  most  probable  value 
of  x.  It  is  evident  that  we  would  obtain  the  same  value  of  x  by  dividing 
the  original  equation  by  6,  so  that  in  the  case  of  a  single  equation  with  a 
single  unknown  quantity  the  most  probable  value  of  that  quantity  is  obtained 
by  simply  solving  the  equation. 


PROBABLE  VALUES  OF  INDEPENDENT  QUANTITIES    275 

Example  2.     Given  the  observation  equations 

2x  =  124.72, 

x  =    62.31, 

7x  =  439.00. 

Multiplying  the  first  equation  by  2,  the  second  by  1,  and  the  third  by  7, 
we  have 

4z  =    249.44; 

x  =      62.31; 

49z  =  3073.00; 

whence  by  addition  we  obtain  the  normal  equation 

54z  =  3384.75, 
the  solution  of  which  gives 

x  =  62.68, 

which  is  hence  the  most  probable  value  that  can  be  obtained  from  the  given 
set  of  observations.  The  student  is  cautioned  against  adding  up  the  obser- 
vation equations  and  solving  for  x,  as  this  plan  does  not  give  the  most 
probable  value  in  such  cases. 

Example  3.     Given  the  observation  equations  «, 

2x  +  y  =  31.65, 
x  -  3y  =  5.03, 
x  -  y  =  11.26. 

Following  the  rule  for  normal  equations,  we  have 

4x  +  2y  =  63.30 
x  —  3y  =  5.03 
x  -  y  =  11.26 

Qx  —  2y  =  79.59  =  normal  equation  in  x; 
and 

2x  +      y  =       31.65 

-  3z  +    9y  =  -  15.09 

-  x  +      y  =  -  11.26 

—  2x  +  \\y  =         5.30  =  normal  equation  in  y. 

It  is  absolutely  essential  in  forming  the  normal  equations  to  multiply  by 
the  algebraic  coefficients  as  illustrated  above,  and  not  simply  by  the  numerical 
value  of  the  coefficient.  Bringing  the  normal  equations  together,  we  have 

Qx  -    2y  =  79.59, 
-  2x  +  Uy  =    5.30. 

Attention  is  called  to  the  fact  that  the  coefficients  in  the  first  row  and  first 
column  are  identical  in  sign,  value,  and  order,  and  that  the  same  is  true  of 
the  second  row  and  second  column.  The  same  law  would  hold  good  if  there 
were  a  third  row  and  a  third  column,  and  so  on  (Art.  162);  and  this  is  a 
check  that  must  never  be  neglected.  Solving  the  two  normal  equations  as 
simultaneous  equations,  we  have 

x  =  14.29        and        y  =  3.08, 
and  these  are  hence  their  most  probable  values. 


276  GEODETIC  SURVEYING 

160.  Indirect  Observations  of  Unequal  Weight  on  Independent 
Quantities.  In  the  case  of  indirect  observations  of  unequal 
weight  on  independent  quantities,  the  most  probable  values  of 
the  unknown  quantities  are  found  by  the  solution  of  one  or  more 
normal  equations  which  involve  the  different  weights  in  their 
formation. 

To  establish  a  rule  for  forming  the  normal  equations  in  the 
case  of  unequal  weights  let  us  re-write  Eqs.  (31),  thus: 


(33) 


+  biy  +  az  ...  —  If i  =  v\  (weight  pi)  } 
a,2X  -\-  b2y  +  C2Z  .  .  .  —  M<2.  =  V2  (weight  pz 


anx  +  bny  +  cnz  .  .  .  -  M n  =  vn  (weight  pn)  J 


In  accordance  with  Art.  156  the   most  probable  values  of  the 
unknown  quantities  are  those  which  give 


.  .  .  +  pnVr?  =  Si  minimum. 

Since  (in  forming  the  normal  equations,  Art.  159)  the  most 
probable  value  of  x  is  desired  for  any  assumed  set  of  values  for 
the  remaining  unknowns,  we  place  the  first  derivative  with 
respect  to  x  equal  to  zero;  whence,  omitting  the  common 
factor  2,  we  have 

dv 


But  from  Eqs.  (33),  under  the  given  assumption  of  fixed  values 
for  all  quantities  excepting  x,  we  obtain 

dvi  dv2 


whence  by  substitution, 

(aipi)vi  +  (a,2p2)v2  .  .  .  +  (anpn)vn  =  0  =  normal  equation  in  x. 
In  a  similar  manner  we  have 

(bipi)vi  +  (b2p2)V2  .  .  .  +  (bnpn)vn  =  0  =  normal  equation  in  y, 

(cipi)vi  +  (C2P2)V2  •  .  .  +  (cnpn)vn  =  0  =  normal  equation  in  2; 

etc.,  etc.; 


PROBABLE  VALUES  OF  INDEPENDENT  QUANTITIES     277 

and  hence  for  forming  the  several  normal  equations  in  the  case 
of  indirect  observations  of  unequal  weight  on  independent  quan- 
tities, we  have  the  following 

RULE  :  To  form  the  normal  equation  for  each  one  of  the  unknown 
quantities,  multiply  each  observation  equation  by  the  product  of  the 
weight  of  that  observation  and  the  algebraic  coefficient  of  that  unknown 
quantity  in  that  equation,  and  add  the  results. 

Having  formed  the  several  normal  equations,  their  solution 
as  simultaneous  equations  gives  the  most  probable  values  of  the 
unknown  quantities. 

Example  1.     Given  the  observation  equations 

3x  =  15°    30'  34"  .6  (weight  2), 
5x  =  25     50  55     .0  (weight  3). 

Multiplying  the  first  equation  by  6  (  =  3  X  2),  and  the  second  equation  by 
15  (  =  5  X  3),  we  have 

l&c  =    93°03'27".6; 
75x  =  387    43  45  .0; 

whence  by  addition  we  obtain  the  normal  equation 

93z  =  480°  47'  12".6, 
the  solution  of  which  gives 

x  =  5°  10'  ll".l, 

which  is  hence  the  most  probable  value  that  can  be  obtained  from  the  given 
set  of  observations. 

Example  2.     Given  the  observation  equations 

x  +    y  =  10.90  (weight  3), 

2x  -    y  =     1.61   (weight  1), 

x  +  3y  =  24.49  (weight  2). 

Following  the  rule  for  normal  equations,  we  have 

3x  +  3y  =  32.70 
4x  -  2y  =  3.22 
2x  +  6y  =  48.98 

9rc  +  7y  =  84.90  =  normal  equation  in  x] 
and 

Sx  +    3y  =    32.70 
-2x  +      y  =-  1.61 
Qx  +  18y  =  146.94 
7x  +  22y  =  178.03  =  normal  equation  in  y. 

Solving  these  two  normal  equations  as  simultaneous,  we  have 

x  =  4.172,         and        y  =  6.765, 
and  these  are  hence  their  most  probable  values. 


278  GEODETIC  SURVEYING 

161.  Reduction  of  Weighted  Observations  to  Equivalent 
Observations  of  Unit  Weight.  To  establish  a  rule  for  this  pur- 
pose let  us  re-  write  Eqs.  (30),  thus: 

a\x  +  biy  -f  c\z  .  .  .  =  MI  (weight  pi), 
+  b2y  +  €22  .  .  .  =  M2  (weight  p-2)  , 


anx  +  bny  +  cnz  .  .  .  =  Mn  (weight  pn). 

Let  C  be  such  a  factor  as  will  change  the  first  of  these  equations 
to  an  equivalent  equation  of  unit  weight,  so  that  we  may  write 

Caix  +  Cbiy  +  Caz  .  .  .  =  CMl     (weight  1), 
d2X  +     bzy  +     C2Z  .  .  .  =     M2     (weight  ^2)  , 

anx  +    bny  +    cnz  .  .  .    =    Mn     (weight  pn)  , 

in  which  the  most  probable  values  of  x,  y,  z,  etc.,  are  to  remain  the 
same  as  in  the  original  equations;  or,  in  other  words,  the  two 
sets  of  equations  are  to  lead  to  the  same  normal  equations.  In 
accordance  with  the  rule  of  Art.  160,  we  have  from  the  first  set 
of  equations 


(34) 


equation 

1  ( etc., etc ) 

and  from  the  second  set  of  equations 


Normal 
equation  | 
in  x 


.  .  .  =C2aiMi) 


-f-  (  .....  etc.,  .........  etc  ........  ) 


(35) 


Comparing  Eq.  (34)  with  Eq.  (35),  term  by  term,  we  find  they  are 
in  all  respects  identical  provided  we  write 


whence 

C  =  V^  ........     (36) 


PROBABLE  VALUES  OF  INDEPENDENT  QUANTITIES     279 

> 

From  the  symmetry  of  the  equations  involved  it  is  evident  that 
the  same  conclusion  would  result  from  a  comparison  of  the  nor- 
mal equations  in  y,  z,  etc.  Hence  it  is  seen  that  an  observation 
equation  of  any  given  weight  may  be  reduced  to  an  equivalent 
equation  of  unit  weight  by  multiplying  the  given  equation  by  the 
square  root  of  the  given  weight.  Evidently  the  converse  of  this 
proposition  is  also  true,  so  that  an  equation  of  unit  weight  can  be 
raised  to  an  equivalent  equation  of  any  given  weight  by  dividing 
the  given  equation  by  the  square  root  of  the  given  weight.  The 
general  laws  of  weights,  as  given  in  Art.  53,  are  readily  derived 
by  an  application  of  these  two  principles.  The  new  equations 
formed  in  the  manner  described,  and  taken  in  conjunction  with 
the  new  weights,  may  be  used  in  any  computations  in  place  of  the 
original  equations,  whenever  so  desired. 

Example  1.     Given  the  observation  equation 
3x  =    8.66  (weight  4). 

What  is  the  equivalent  observation  equation  of  unit  weight? 
Since  the  square  root  of  4  is  2,  we  have 

Qx  =  17.32  (weight  1) 
as  the  equivalent  equation. 

Example  2.     Given  the  observation  equation 
3x  +  Qy  =  11.04  (weight  1). 

What  is  the  equivalent  observation  equation  of  the  weight  9? 
Since  the  square  root  of  9  is  3,  we  have 

x  +  2y  =    3.68  (weight  9) 
as  the  equivalent  equation. 

Example  3.     Given  the  observation  equation 

x  +  y  —  2z  =  a  (weight  3) . 

What  is  the  equivalent  observation  equation  of  the  weight  7? 
Multiplying  by  Vs  and  dividing  by  VT,  we  have 

V$x  +  V?  y  -  2V?  z  =  V?  a  (weight  7) 
as  the  equivalent  equation. 


280  GEODETIC  SURVEYING 

162.  Law  of  the  Coefficients  in  Normal  Equations.  In  accord- 
ance with  Art.  158,  we  may  write  in  general  for  any  set  of 
obser  vations 

a\x  +  b\y  +c\z  .  .  .  =  M\    (weight  pi), 
a,2X  +  b2y  +  c2z  .  .  .  =  M2   (weight  p2)  , 

anx  +  bny  +  cnz.  .  .  =  Mn  (weight  pn). 

Forming  the  normal  equation  in  x  in  accordance  with  the  rule  of 
Art.  160,  the  multiplying  factors  are  piai,  p2d2)  etc.,  giving 


Pidi2x  + 

p2d2c2z  .  .  .  =  p2a2M2 


p,tdn2x  +  pnanbny 


=normal  equation  inx. 

Similarly,  for  the  normal  equation  in  y,  the  multiplying  factors 
are  pi&i,  p2b2,  etc.,  giving 

I^(pab)x  +  ^(pb2)y  +  ^(pbc)z  .  .  .=2(p6M)  =  normal  equation  my. 

Similarly,  for  the  normal  equation  in  z,  the  multiplying  factors 
are  pid,  p2c2}  etc.,  giving 


S  (pac}x  +  2  (pbc)  y  +  2  (pc2)z  .  .  .  =  2  (pcM)  =  normal  equation  in  z  • 

and  so  on  for  any  additional  unknown  quantities.     Collecting 
the  several  normal  equations  together,  we  have 


etc.,  etc. 

An  examination  of  these  equations  shows  that  the  coefficients  in 
the  first  row  and  in  the  first  column  are  identical  in  sign,  value, 
and  order.  The  same  proposition  is  true  of  the  second  row  and 
second  column,  the  third  row  and  third  column,  and  so  on.  This 
is  hence  the  general  law  of  the  coefficients  in  any  set  of  normal 
equations,  and  furnishes  a  check  on  the  work  that  should  never 
be  neglected. 


PROBABLE  VALUES  OF  INDEPENDENT  QUANTITIES    281 

Example.     Let  the  following  observation  equations  be  given: 

2x  -    z  =  8.71  (weight  2), 

x  -  2y  +  3z  =  2.16  (weight  1), 

y  -  2z  =  1.07  (weight  2), 

x  -  3y  =  1.93  (weight  1). 

The  corresponding  normal  equations  are 

10z  —    5y  —      z  =       38.93  =  normal  equation  in  x; 

—  5x  +  15y  —  lOz  =  —    7.97  =  normal  equation  in  y; 

—  x  —  Wy  +  192  =  —  15.22  =  normal  equation  in  z; 

from  which  we  have 

/  First  row  are  +  10,   —    5,    —    1. 

Coefficients  in  > 


-,     „  •     .    • 
Coefficients  in 


ru--.  .      ,    .         Third  row  are  -     1,    -  10,   +  19. 

Coefficients  in 


/  Second  row  are         —    5,   +  15,    —  13. 
[  gecond  column  are  _    5>   +  15j   _  1(X 


163.  Reduced  Observation  Equations.    Such  observation  equa- 
tions as  are  likely  to  occur  in  geodetic  work  may  be  written  under 
the  general  form 

ax  +  by  +  cz  +  etc.  =  M.     .     .     .     .     .     (37) 

Substituting 

1 


X  =  Xi          VL 

y  =  y\  +  v2 
z  =  zi  +  v3 


(38) 


in  which  Xi,  y\,  z\,  etc.,  are  any  assumed  constants,  and  vi,  V2, 
etc.,  are  new  unknowns,  the  equation  takes  the  reduced  form 


avi  +  bv2  +  cv$  +  etc.  =  M  —  (axi  +  byi  +  czi  +  etc.).    (39) 

In  this  new  equation  it  will  be  noticed  that  the  first  member  is 
identical  in  form  with  the  first  member  of  the  original  equation, 
the  only  change  being  the  substitution  of  the  new  variables  for 
the  old  ones;  and  that  the  second  member  is  what  the  original 
equation  reduces  to  when  the  assumed  constants  are  substituted 
for  the  corresponding  variables.  The  reduced  observation 
Eq.  (39)  may  therefore  be  written  out  at  once  from  the  observa- 


282  GEODETIC  SURVEYING 

tion  Eq.  (37),  without  going  through  the  direct  substitution  of 
Eqs.  (38).  Particular  attention  is  called  to  the  second  member 
of  Eq.  (39),  in  which  it  is  seen  that  the  result  due  in  any  case  to 
the  use  of  the  assumed  values  of  x,  y,  etc.,  must  always  be  sub- 
tracted from  the  corresponding  measured  value,  and  not  vice 
versa,  as  any  error  in  sign  will  render  the  whole  computation 
worthless.  It  is  also  to  be  noted  that  the  original  weights  apply 
also  to  the  reduced  observation  equations,  since  these  are  simply 
different  expressions  for  the  original  equations. 

In  view  of  the  meaning  of  the  terms  in  Eqs.  (38)  it  is  evident 
that  the  most  probable  value  of  x  is  that  which  is  due  to  the  most 
probable  value  of  vi,  and  correspondingly  with  all  the  other 
unknowns.  We  may,  therefore,  in  any  case,  reduce  all  the 
original  observation  equations  to  the  form  of  Eq.  (39),  determine 
from  these  reduced  equations  the  most  probable  values  of  wi,  V2, 
etc.,  and  then  by  means  of  Eqs.  (38)  determine  the  most  probable 
values  of  x,  y,  z,  etc.  The  object  of  this  method  of  computation 
is  to  save  labor  by  keeping  all  the  work  in  small  numbers.  This 
result  is  accomplished  by  assigning  to  x\,  yi,  etc.,  values  which 
are  known  to  be  approximately  equal  to  x,  y,  etc.,  as  this  will 
evidently  reduce  the  second  term  of  equations  like  Eq.  (39)  to 
values  approximating  zero.  Approximate  values  of  the  unknowns 
are  always  obtainable  from  an  inspection  of  the  observation 
equations,  or  by  obvious  combinations  thereof. 

Example  1.     Given  the  following  observation  equations: 

x  =  178.651, 

y  =  204.196, 

x  +  y  =  382.859, 

2x  +  y  =  561.522; 

to  find  the  most  probable  values  of  the  unknowns  by  the  method  of  reduced 
observation  equations. 

Assuming  for  the  most  probable  values 

x  =  178.651  +  Vi, 
y  =  204.196  +  v2, 

we  have  by  substitution  in  the  observation  equations,  or  directly  in  accord- 
ance with  Eq.  (39), 

Vi  =  0.000; 

v2  =  0.000; 

Vl  +  v2  =  0.012; 

2K  +  1    =0.024. 


PROBABLE  VALUES  OF  INDEPENDENT  QUANTITIES    283 

> 

Forming  the  normal  equations  frona<  these  reduced  observation  equations, 
we  have 

6vi  +  3v2  =  0.060; 

3vi  +  3v2  =  0.036; 
whose  solution  gives 

vi  =  0.008        and        v«  =  0.004; 

whence  for  the  most  probable  values  of  x  and  y  we  have 

x  =  178.651  +  0.008  =  178.659; 
y  =  204.196  +  0.004  =  204.200. 

These  results  are  identical  with  what  would  have  been  obtained  if  any  other 
values  had  been  used  for  x\  and  y\,  or  if  the  normal  equations  had  been 
formed  directly  from  the  original  observation  equations. 

Example  2.     Given  the  following  observation  equations: 

2x  +    y  =  116°  38'  19".7  (weight  2), 

x  +    y  =    73     17  22   .1  (weight  1), 

x  -    y  =    13    24  28  .3  (weight  3), 

x  +  2y  =  103     13  47  .7  (weight  1); 

to  find  the  most  probable  values  of  the  unknowns  by  the  method  of  reduced 
observation  equations. 

It  is  readily  seen  that  the  first  two  of  these  equations  are  exactly  satisfied 
if  we  write 

x  =  43°  20'  57".6; 

y  =  29    56    24  .5. 

Adopting  these  as  the  approximate  values  we  have  for  the  most  probable 
values 

x  =  43°  20'  57".6  +  Vl; 

y  =  29    56    24  .5  +  v2; 

whence  by  substitution  in  the  observation  equations,  or  directly  in  accord- 
ance with  Eq.  (39),  we  have 

2vi  +    v2  =  0".0  (weight  2); 

Vl  +    V2  =  0  .0  (weight  1); 

vi  —    v2  =  -  4  .8  (weight  3) ; 

vi  +  2v2  =  1  .1  (weight  1). 

Forming  the  normal  equations  from  these  reduced  observation  equations, 
we  have 

130i  +    4v2  =  -  13".3; 
4*1  +  10v2  =       16  .6; 
whose  solution  gives 

vj.  =  -  1".75        and        v2  =  +  2".36; 

whence  for  the  most  probable  values  of  x  and  y  we  have 

x  =  (43°  20'  57".6)  -  1".75  =  43°  20'  55".85; 
y  =  (29    56    24  .5)  +2  .36  =  29    56    26  .86. 

As  in  the  previous  example  these  results  are  identical  with  what  would  have 
been  obtained  if  any  other  values  had  been  used  for  x\  and  y\,  or  if  the  normal 
equations  had  been  formed  directly  from  the  original  observation  equations. 


CHAPTER  XII 

MOST  PROBABLE  VALUES   OF  CONDITIONED  AND  COMPUTED 

QUANTITIES 

164.  Conditional  Equations.  The  methods  heretofore  given 
determine  the  most  probable  values  in  all  cases  where  the  quanti- 
ties observed  are  independent  of  each  other.  In  many  cases,  how- 
ever, certain  rigorous  conditions  must  also  be  satisfied,  so  that  any 
change  in  one  quantity  demands  an  equivalent  change  in  one 
or  more  other  quantities.  Thus  in  a  triangle  the  three  angles 
can  not  have  independent  values,  but  only  such  values  as  will  add 
up  to  exactly  180°.  When  quantities  are  thus  dependent  on  each 
other  they  are  called  conditioned  quantities.  By  an  equation  of 
condition  or  a  conditional  equation  is  meant  an  equation  which 
expresses  a  relation  that  must  exist  among  dependent  quantities. 
Thus  if  x,  y,  and  z  denote  the  three  angles  of  a  triangle  we  have 
the  corresponding  conditional  equation 

x  +  y  +  z  =  180°. 

In  such  a  case  the  most  probable  values  of  x,  y,  and  z  are  not 
those  values  which  may  be  individually  the  most  probable,  but 
those  values  which  belong  to  the  most  probable  set  of  values  that 
will  satisfy  the  given  conditional  equation.  In  accordance  with 
the  principles  heretofore  established  that  set  of  values  is  the  most 
probable  which  leads  to  a  minimum  value  for  the  sum  of  the 
weighted  squares  of  the  resulting  residuals  in  the  observation 
equations. 

In  the  problems  which  occur  in  geodetic  work  the  conditional 
equations  may  in  general  be  expressed  in  the  form 

aix  +  a2y  .  .  .  +  aut    =  Ea 


wi\x  -\-  ni2y  .  .  .  ~h  mut  =  ET 


284 


PROBABLE  VALUES  OF  CONDITIONED  QUANTITIES     285 

in  which  x,  y,  t,  etc.,  are  the  most  probable  values  of  the  unknown 
quantities,  and  u  is  the  number  of  such  quantities.  It  is  evident 
that  the  number  of  independent  conditional  equations  must  be 
less  than  the  number  of  unknown  quantities.  For  if  these  equa- 
tions are  equal  in  number  with  the  unknown  quantities  their 
solution  as  simultaneous  equations  will  determine  absolute  values 
for  the  unknowns,  so  that  such  quantities  can  not  be  the  subject 
of  measurement.  While  if  the  number  of  these  equations  exceeds 
the  number  of  unknowns,  such  equations  can  not  all  be  inde- 
pendent without  some  of  them  being  inconsistent.  On  the  other 
hand  the  total  number  of  equations  (sum  of  the  observation  and 
the  independent  conditional  equations)  must  exceed  the  number 
of  unknown  quantities.  For  if  the  total  number  of  equations  is 
equal  to  the  number  of  unknown  quantities,  their  solution  as 
simultaneous  equations  will  furnish  a  set  of  values  which  will 
exactly  satisfy  all  the  equations,  without  involving  any  question 
of  what  values  may  be  the  most  probable.  While  if  the  total 
number  of  equations  is  less  than  the  number  of  unknown 
quantities  the  problem  becomes  indeterminate. 

There  are  in  general  two  methods  of  finding  the  most  probable 
values  of  the  unknown  quantities  in  cases  involving  conditioned 
quantities.  In  the  first  method  the  conditional  equations  are 
avoided  (or  eliminated)  by  impressing  their  significance  on  the 
observation  equations,  which  reduces  the  problem  to  the  cases 
previously  given.  In  the  second  method  the  observation  equa- 
tions are  eliminated  by  impressing  their  significance  on  the  con- 
ditional equations,  when  the  solution  may  be  effected  by  the 
method  of  correlatives  (Art.  167).  The  first  method  is  the  most 
direct  in  elementary  p  oblems,  but  the  second  method  greatly 
reduces  the  work  of  computation  in  the  case  of  complicated 
problems. 

165.  Avoidance  of  Conditional  Equations.  In  a  large  num- 
ber of  problems  it  is  possible  to  avoid  the  use  of  conditional 
equations  by  the  manner  in  which  the  observation  equations  are 
expressed.  The  conditions  which  have  to  be  satisfied  in  any 
given  case  are  never  alone  sufficient  to  determine  the  values  of 
any  of  the  unknown  quantities,  as  otherwise  these  quantities 
would  not  be  the  subject  of  observation.  It  is  only  after  definite 
values  have  been  assigned  to  some  of  the  unknown  quantities 
that  the  conditional  equations  limit  the  values  of  the  remaining 


286  GEODETIC  SURVEYING 

ones.  In  any  problem,  therefore,  a  certain  number  of  values 
may  be  regarded  as  independent  of  the  conditional  equations, 
whence  the  remaining  values  become  dependent  on  the  independent 
ones.  Thus  in  a  triangle  any  two  of  the  angles  may  be  regarded 
as  independent,  whence  the  remaining  one  becomes  dependent 
on  these  two,  since  the  total  sum  must  be  180°.  In  any  elementary 
problem  it  is  generally  self  evident  as  to  how  many  quantities 
must  be  regarded  as  independent,  and  which  ones  may  be  so  taken. 
In  such  cases  the  conditional  equations  may  be  avoided  by 
writing  out  all  of  the  observation  equations  in  terms  of  the 
independent  quantities.  The  most  probable  values  of  these 
quantities  may  then  be  found  by  the  regular  rules  for  independent 
quantities,  whence  the  most  probable  values  of  the  remaining 
quantities  are  determined  by  the  surrounding  conditions  that 
must  be  satisfied. 

Example  1.     Referring  to  Fig.  65,  the  following  angular  measurements 
have  been  made: 

x  =  28°  11'  52".2; 
y  =  30  42  22  .7; 
z  =  58  54  17  .6. 

What  are  the  most  probable  values  of  these  angles? 
It  is  evident  from  the  figure  that  these  angles  are  sub- 
ject to  the  condition 


x  +  y  =  z. 

•wever,  \ 
form 


IG'      *  If,  however,  we  write  the  observation  equations  in  the 


x  =  28°  11'  52".2; 

y  =  30    42    22  .7; 

x  +  y  =  58    54    17  .6; 

the  conditional  equation  is  avoided,  since  x  and  y  are  manifestly  inde- 
pendent angles.  The  second  set  of  observation  equations  must  lead  to 
exactly  the  same  figures  for  the  most  probable  values  of  x  and  y  (and  hence 
for  z)  as  the  first  set,  since  it  is  only  another  way  of  stating  exactly  the 
same  thing.  Since  x  and  y  are  independent  angles  we  may  write  for  the 
most  probable  values 

x  =  28°  11'  52".2  +  vi; 

y  =  30    42    22  .7  +  v2; 

whence  the  reduced  observation  equations  are 

t>i  =  0".0; 

v2  =  0  .0; 

Vi  +  vz  =  2  .7. 


PROBABLE  VALUES  OF  CONDITIONED  QUANTITIES     287 

i 

The  corresponding  normal  equations,  are 

2*  +    v2  =  2".7; 
»i  +  2y2  =  2  .7; 
whose  solution  gives 

vi  =  +  0".9     and     v2  =  +  0".9. 
The  most  probable  values  of  the  given  angles  are  therefore 

x  =  28°  11'  53".l; 
y  =  30  42  23  .6; 
z  =  58  54  16  .7. 

Example  2.     Referring  to  Fig.  66,  the  following  angular  measurements 
have  been  made: 

x  =  80°  45'  37".6  (weight  2); 
y  =  135  08  14  .9  (weight  1); 
z  =  144  06  10  .8  (weight  3). 

What  are  the  most  probable  values  of  these  angles? 
It  is   evident  from  the  figure  that  these  angles  are 
subject  to  the  condition 

x  +  y  +  z  =  360°. 

Any  two  angles  at  a  point,  such  as  x  and  y,  may 
be  regarded  as  independent,  so  that  the  conditional 
equation  is  avoided  by  writing  all  the  observation 
equations  in  terms  of  these  two  quantities.  Thus  we  FIG.  66. 

write: 

x  =    80°  45'  37".6  (weight  2); 

y  =  135    08    14  .9  (weight  1); 

360°  -  (x  +  y)  =  144    06    10  .8  (weight  3) ; 

whence  by  substituting 

x  =    80°  45'  37".6  +  vlt 
y  =  135    08    14  .9  +  v2, 
we  have 

vi  =       0".0  (weight  2); 

v2  =       0  .0  (weight  1); 

vi  +  v2  =  —  3  .3  (weight  3) ; 

from  which  the  normal  equations  are 

5vi  +  Zv2  =  -  9".9; 
30!  +  4r2  =  -  9  .9; 
whose  solution  gives 

V!  =  -  0".9     and    v2  =  -  1".8. 
The  most  probable  values  of  the  given  angles  are  therefore 

x  =  80°  45'  36".7; 
y  =  135  08  13  .1; 
z  =  144  06  10  .2. 


288  GEODETIC  SURVEYING 

166.  Elimination  of  Conditional  Equations.  If  the  con- 
ditional equations  can  not  be  directly  avoided,  as  in  the 
preceding  article,  the  same  result  may  be  indirectly  accomplished 
by  algebraic  elimination,  as  about  to  be  explained.  The  number 
of  unknown  quantities  (Art.  164)  necessarily  exceeds  the  number 
of  independent  conditional  equations.  The  number  of  dependent 
unknowns,  however,  can  not  exceed  the  number  of  independent 
conditional  equations,  since  any  values  whatever  may  be  assigned 
to  the  remaining  unknowns  and  still  leave  the  equations  capable 
of  solution.  Thus  if  there  are  five  unknowns  and  three  independent 
conditional  equations,  any  values  may  be  assigned  to  any  two  of 
the  unknowns,  leaving  three  equations  with  three  unknowns  and 
hence  capable  of  solution.  The  unknowns  selected  as  arbitrary 
values  thus  become  independent  quantities  on  which  all  the  others 
must  depend,  and  the  number  of  unknowns  which  may  be  thus 
selected  as  independent  quantities  is  evidently  equal  to  the 
difference  between  the  total  number  of  unknowns  and  the  number 
of  independent  conditional  equations.  If  the  most  probable 
values  are  assigned  to  the  independent  quantities,  the  most 
probable  values  of  the  dependent  quantities  then  become  known 
by  substituting  the  values  of  the  independent  quantities  in  the 
dependent  equations.  The  general  plan  of  procedure  is  as 
follows: 

1.  Determine  the  number  of  independent  unknowns  by  sub- 
tracting the  number  of  conditional  equations  from  the  number 
of  unknown  quantities. 

2.  Select  this  number  of  unknowns  as  independent  quantities. 

3.  Transpose  the  conditional  equations  so  that  the  dependent 
quantities  are  all  on  the  left-hand  side  and  the  independent  quan- 
tities on  the  right-hand  side. 

4.  Solve  the  conditional  equations  for  the  dependent  unknowns, 
which  will  thus  express  each  of  these  dependent  unknowns  in 
terms  of  the  independent  unknowns. 

5.  Substitute  these  values  of  the  dependent  unknowns  in  the 
observation   equations,    which   will   then    contain   nothing   but 
independent  unknowns. 

6.  Find  the  most  probable  values  of  the  independent  unknowns 
from  these  modified  observation  equations  by  the  regular  rules 
for  independent  quantities. 

7.  Substitute  these  values  of  the  independent  unknowns  in 


PROBABLE  VALUES  OF  CONDITIONED  QUANTITIES    289 

the  expressions  for  the  dependent  unknowns,  and  thus  determine 
the  most  probable  values  of  the  remaining  quantities. 

Example,     Given  the  following  data,  to  find  the  most  probable  values 
of  x,  y,  and  z: 

[  x  =  17.82  (weight  2); 

Observation  equations  i  y  =  15.11  (weight  4); 
I  z  =  29.16  (weight  3). 

Conditional  equations     {  £  +  *  _  ^  -  112.00; 

The  solution  is  as  follows: 

Number  of  unknown  quantities  =  3. 
Number  of  conditional  equations  =  2. 
Number  of  independent  quantities  =  1. 

Let  x  be  the  independent  quantity,  and  y  and  z  the  dependent  quantities. 
Transpose  the  conditional  equations  so  as  to  leave  only  the  dependent 
quantities  on  the  left  hand  side,  thus: 

5y         =  112.00  -  2z; 
y  —  z  =    39.00  —  3x. 

Solve  for  the  dependent  quantities,  giving  the  dependent  equations 

y  =       22 .40  -  0.4z; 
z  =  -  16.60  +  2.6x. 

Substitute  in  the  observation  equations,  giving 

x  =  17.82  (weight  2); 
22.40  -  OAx  =  15.11  (weight  4); 
-  16.60  +  2.6z  =  29.16  (weight  3); 
whence 

x  =  17.82  (weight  2); 
0.4z  =  7.29  (weight  4); 
2.6z  =  45.76  (weights); 

in  which  x  is  an  independent  unknown.  Forming  the  normal  equation 
by  multiplying  the  above  equations  respectively  by  2,  1.6,  and  7.8,  we  have 

2.00z  =  35.640, 

0.64z  =  11.664, 

20.28z  =  356.928 


22.92z  =  404.232; 
x  =  17.637; 

which,  substituted  in  the  first  dependent  equation,  gives, 
y  =  22.40  -  0.4(17.637)  =  15.345, 


290  GEODETIC  SURVEYING 

and  substituted  in  the  second  dependent  equation,  gives 

z  =  -  16.60  +  2.6(17.637)  =  29.255; 
so  that  for  the  most  probable  values  of  the  unknown  quantities,  we  have 

x  =  17.637; 
y  =  15.345; 
z  =  29.255. 

As  a  check  on  the  work  of  computation,  we  may  substitute  these  values  in 
the  conditional  equations,  giving 

2x  +  5y      •    =  35.274  +  76.725  =  111.999; 

3x  +    y  -  z  =  52.911  +  15.345  -  29.255  =    39.001; 

from  which  it  is  seen  that  each  equation  checks  with  the  corresponding 
conditional  equation  within  0.001,  which  is  an  entirely  satisfactory  check. 
The  essential  feature  of  the  above  method  is  the  elimination  of  the  con- 
ditional equations.  In  Art.  167  the  same  problem  is  worked  out  by  elim- 
inating the  observation  equations.  The  results  obtained  are  of  course 
identical. 

167.  Method  of  Correlatives.  The  general  method  of  correla- 
tives is  beyond  the  scope  of  the  present  volume.  The  case  here 
given  is  the  only  one  that  is  likely  to  be  of  service  to  the  civil 
engineer.  In  this  case  the  observations  are  made  directly  on 
each  unknown  quantity,  and  the  number  of  observation  equations 
equals  the  number  of  unknown  quantities.  Let  u  be  the  number 
of  unknown  quantities,  for  which  the  observation  equations  may 
be  written 

x  =  M i     (weight  pi); 

y  =  M2     (weight  p2) ; 

t  =  Mu     (weight  pj; 

and  for  which  (Art.  164)  the  conditional  equations  may  be  written 
aix  +  azy  .  .  .  +  aut    =  Ea 
bix  +  b2y  .  .  .  +  W   =  Eb 


mix  +  m2y  .  .  .  +  mut  =  Er 


(41) 


If,  as  heretofore,  x,  y,  t,  etc.,  be  understood  to  represent  the  most 
probable  values  of  the  unknown  quantities,  and  vi,  V2,  vu,  etc., 


PROBABLE  VALUES  OF  CONDITIONED  QUANTITIES    291 


represent  the  corresponding  residuals  in  the  given  equations,  we 
may  write 

x  =  M i  +  Vi     (weight  pi) 

y  =  M2  +  v2     (weight  p2) 


t  =  Mu  +  vu     (weight  pu)  J 

which,  substituted  in  Eqs.  (41),  give  the  conditional  equations 
+  a2^2  .  .  .  +  auvu  =  Ea  - 
+  b2V2   .  .  .  +  uuvu    =  &b 


buvu    =  E    -  26M 


m2v2  .  .  .  -f  muvu  =  Em  — 


(43) 


As  explained  in  Art.  164,  these  conditional  equations  must  be 
less  in  number  than  the  number  of  unknown  quantities.  The 
values  of  vi,  v2,  etc.,  thus  become  indeterminate,  and  an  infinite 
number  of  sets  of  values  will  satisfy  the  equations.  The  values 
in  any  one  set  (called  simultaneous  values)  are  not  independent, 
however,  as  they  must  be  such  as  will  satisfy  the  above  equations. 
If  vi,  V2,  etc.,  in  Eqs.  (43)  are  assumed  to  vary  through  all 
possible  simultaneous  values  due  to  any  set  of  values  dvi,  dv2, 
etc.,  and  all  possible  sets  of  values  dvi,  dv2,  etc.,  are  taken  in  turn, 
the  most  probable  set  of  values  vit  v2,  etc.,  for  the  given  set  of 
observations  will  eventually  be  reached.  The  values  dvi,  dv2, 
etc.,  in  any  one  set,  however,  can  not  be  independent,  as  it  is 
evident  that  dependent  quantities  can  not  be  varied  indepen- 
dently. Differentiating  Eqs.  (43),  we  have 


+  a2dv2  ...  .  +  audvu   =  0 


budvu 


0 


0 


(44) 


and  these  new  equations  of  condition  show  the  relations  that  must 
exist  among  the  quantities  dvi,  dv2,  etc.  Since  the  number  of 
equations  is  less  than  the  number  of  quantities  dvi,  dv2,  etc.,  it 
follows  that  an  infinite  number  of  sets  of  simultaneous  values  of 
dvi,  dv2,  etc.,  is  possible.  In  order  to  involve  Eqs.  (44)  simul- 


292 


GEODETIC  SUKVEYING 


taneously  in  an  algebraic  discussion  it  is  necessary  to  replace 
them  by  a  single  equivalent  equation,  meaning  an  equation  so 
formed  that  the  only  values  which  will  satisfy  it  are  those  which 
will  individually  satisfy  the  original  equations  which  it  replaces. 
This  is  done  by  writing 


a2dv2 


m2dv2 


audvu) 


budvu) 


mudvu) 


=  0; 


(45) 


in  which  /bi,  £2,  etc.,  are  independent  constants  which  may  have 
any  possible  values  assigned  to  them  at  pleasure.  Since  Eq.  (45) 
must  by  agreement  remain  true  for  all  possible  sets  of  values 
kij  k2)  etc.,  its  component  members  must  individually  remain 
equal  to  zero.  But  these  component  members  are  identical  with 
the  first  members  of  the  original  conditional  equations,  so  that 
no  set  of  values  dvi,  dv2,  etc.,  can  satisfy  Eq.  (45)  unless  it  can 
also  satisfy  each  of  Eqs.  (44).  The  values  in  any  such  set  are 
called  simultaneous  values. 

In  order  to  determine  the  most  probable  values  of  vi,  v2,  etc., 
we  must  have  (Art.  156) 

Pivi2  +  p2v22  .  .  .  +  puvu2  =  a  minimum. 

In  accordance  with  the  principles  of  the  calculus  for  the  case  of 
dependent  quantities  the  first  derivative  of  this  expression  must 
equal  zero  for  every  possible  set  of  values  dv\,  dv2,  etc.  Hence, 
by  differentiating,  and  omitting  the  factor  2,  we  have 

Pividvi  +  p2v2dv2  .  .  .  +  puvudvu  =  0,    .     .     .     (46) 

in  which  dvi,  dv2,  etc.,  must  be  simultaneous  values.  Since  these 
values  are  also  simultaneous  in  Eq.  (45),  we  may  combine  this 
equation  with  Eq.  (46)  and  write 


p2v2dv2  .  .  .  +  Puvudv 


a2dv2  . 
b2dv2    . 


m2dv2  .  .  .  +  mudvu)    . 


PROBABLE  VALUES  OF  CONDITIONED  QUANTITIES    293 
whence,  by  rearranging  the  te#ns,  we  have 


[puvu  — 


k2b2  .  .  .  +  kmm2)]dv2 


k2bu  .  .  .  +  kmmu)]dvu  J 


,_ 


Since  ki,  k2)  etc.,  are  independent  and  arbitrary  constants,  it  is 
evident  that  this  equation  can  not  be  true  unless  its  component 
members  are  each  equal  to  zero,  so  that 


+  k2bi  •  •  •  +  kmmi)]dvi  =  0; 
etc.; 


etc., 
from  which  we  have 


p2v2  = 


+  /C2&1 
+  k2b2 

4-    fcofc 

...  +  kmmi  ' 
.  .  .  +  &mm2 

-1-  k...m. 

(48) 


as  the  general  equations  of  condition   for    the   most  probable 
values  of  vi,  v2,  etc. 

It  is  evident  that  Eqs.  (48)  can  not  be  solved  for  v\,  v2,  etc., 
until  definite  values  have  been  assigned  to  kit  k2,  etc.  In  the 
general  discussion  of  the  problem  the  values  of  &i,  k2,  etc.,  have 
been  entirely  arbitrary,  since  the  numerical  requirements  of 
Eqs.  (43)  vanished  in  the  differentiation.  In  any  particular  case, 
however,  the  m  conditional  Eqs.  (43)  must  be  numerically  satisfied 
in  order  to  satisfy  the  rigid  geometrical  conditions  of  the  case, 
while  the  u  conditional  Eqs.  (48)  must  be  satisfied  in  order  to  have 
the  most  probable  values  for  v\,  v2,  etc.  There  are  thus  m  -f-  u 
simultaneous  equations  to  be  satisfied.  But  there  are  also  m  +  u 
unknown  quantities,  since  the  m  unknown  quatities  ki,  k2,  etc., 
corresponding  to  the  m  conditional  Eqs.  (43),  have  been  added 
to  the  u  unknown  quantities  vi,  v2,  etc.  In  any  particular  case, 
therefore,  there  is  but  one  set  of  values  for  the  m  unknown  quan- 
tities kij  k2,  etc.,  and  the  u  unknown  quantities  vi,  v2)  etc.,  that 
will  satisfy  the  m  +  u  equations  consisting  of  Eqs.  (43)  and  (48). 
The  auxiliary  quantities  ki,  k2)  etc.,  are  called  the  correlatives 


294 


GEODETIC  SURVEYING 


(or  correlates)  of  the  corresponding  conditional  Eqs.  (43),  and  the 
quantities  v\,  V2,  etc.,  are  the  most  probable  values  of  the  residual 
errors  in  the  observation  equations.  Substituting  in  Eqs.  (43) 
the  values  of  !>i,  V2,  etc.,  due  to  Eqs.  (48),  we  have 


p 

V 
¥2  + 


p 

~p 

w 
p 


—  =  Eb  — 
p 


=  E     - 


(49) 


in  which 


P  Pi 

etc., 


P2 


etc. 


Attention  is  called  to  the  fact  that  the  law  of  the  coefficients 
in  Eqs.  (49)  is  the  same  (Art.  162)  as  the  law  of  the  coefficients 
for  normal  equations,  and  this  is  a  check  that  should  never  be 
neglected.  It  is  evident  that  &i,  £2,  etc.,  can  be  found  by  solving 
the  simultaneous  Eqs.  (49).  Then,  from  Eqs.  (48),  we  have 


Vl  = 


Pi          Pi 


p2       p2  mp2 


bu 


;    ...    (50) 


and  from  Eqs.  (42), 


x  =  MI 

y  =  M2 


t  =  Mu  +  v 


(51) 


PROBABLE  VALUES  OF  CONDITIONED  QUANTITIES    295 


in  which  x,  y,  t,  etc.,  are  the  moist  probable  values  of  the  quantities 
whose  observed  values  were  MI,  M^  Mw  etc. 

Example.     Given  the  following  data,  to  find  the  most  probable  values  of 
x,  y,  and  «•: 

x  =  17.82     (weight  2); 

Observation  equations    y  =  15.11     (weight  4); 
z  =  29.16     (weight  3). 


'  •   Conditional  equations  {£ 

In  this  case  we  have 


_  , 


Ea 

=  112.00 

#6 

39.00 

SaM  =  111.19 

26M  = 

39.41 

Ea 

-  S< 

TiVz    -— 

0.81 

Eb  -  26M  =  - 

-    0.41 

Ifl 

=  17 

,82 

ax  =  2 

61  =  3 

Pi  =  2 

MZ 

=  15, 

11 

c^  =  5 

62  =  1 

P2    =    4 

M3 

=  29. 

16 

az  =  0 

6S  =  -1 

PS  =  3 

sfl2 

_  33 

ai  _  i 

61  _  3 

P 

4 

Pi 

Pi       2 

va& 

17 

a2       5 

62       1 

"p 

=  T 

P2           4 

P2           4 

2- 

_61 

2?  =0 

63  _ 

1 

P 

12 

P3 

3 

33, 


We  thus  have 


0.81 


§*,;,- cw, 


r  fei  =  +  0.2454. 
fo  =  -  0.2859. 


vi  =  0.2454  X  1  -  0.2859  X  I  =  -  0.183; 
v2  =  0.2454  Xf-  0.2859  X  1  =  +  0.235; 
vs  =  0.2454  X  0  +  0.2859  X  i  =  +  0.095; 

whence,  for  the  most  probable  values  of  x,  y,  and  z,  we  have 

x  =  Mi  +  «>i  =  17.82  -  0.183  =  17.637; 
y  =  M2  +  v2  =  15.11  +  0.235  =  15.345; 
z  =  Ms  +  v3  =  29.16  +  0.095  =  29.255. 

As  a  check  on  the  work  of  computation  we  may  substitute  these  values  in 
the  conditional  equations,  giving 

2x  +  5y         =  35.274  +  76.725  =  111.999; 

3x  +    y  -  z  =  52.911  +  15.345  -  29.255  =    39.001; 

from  which  it  is  seen  that  each  equation  checks  with  the  corresponding  con- 
ditional equation  within  0.001,  which  is  an  entirely  satisfactory  check.     The 


296  GEODETIC  SURVEYING 

essential  feature  of  the  above  method  is  the  elimination  of  the  observation 
equations.  In  Art.  166  the  same  problem  is  worked  out  by  eliminating  the 
conditional  equations.  The  results  obtained  are  of  course  identical. 

168.  Most  Probable  Values  of  Computed  Quantities.  By  a  com- 
puted quantity  is  meant  a  value  derived  from  one  or  more  observed 
quantities  by  means  of  some  geometric  or  analytic  relation. 
The  most  probable  values  of  computed  quantities  are  found  from 
the  most  probable  values  of  the  observed  quantities  by  employ- 
ing the  same  rules  that  are  used  with  mathematically  exact  quan- 
tities. Thus  the  most  probable  value  of  the  area  of  a  rectangle 
is  that  which  is  given  by  the  product  of  the  most  probable  values 
of  its  base  and  altitude;  the  most  probable  value  of  the  circum- 
ference of  a  circle  is  equal  to  x  times  the  most  probable  value  of 
its  diameter;  and  so  on. 


CHAPTER  XIII 

PROBABLE   ERRORS    OF   OBSERVED   AND  COMPUTED  QUANTITIES 

A.     OF  OBSERVED  QUANTITIES 

169.  General  Considerations.     The  most  probable  value  of 
a  quantity  does  not  in  itself  convey  any  idea  of  the  precision  of 
the  determination,  nor  of  the  favorable  or  unfavorable  circum- 
stances surrounding  the  individual  measurements.     Any  single 
measurement  tends  to  lie  closer  to  the  truth  the  finer  the  instru- 
ment and  the  method  used,  the  greater  the  skill  of  the  observer, 
the  better  the  atmospheric  conditions,  etc.     The  accidental  errors 
of  observation  tend  to  be  more  thoroughly  eliminated  from  the 
average  value  of  a  series  of  measurements  the  greater  the  number 
of  measurements  which  are  averaged  together.     Some  criterion 
or  standard  of  judgment  is  therefore  necessary  as  a  gage  of  pre- 
cision.    Since  the  probability  curve  for  any  particular  case  shows 
the  facility  of  error  in  that  case,  and  thus  represents  all  the  sur- 
rounding  circumstances    under    which   the    given    observations 
were  taken,  it  is  evident  that  some  suitable  function  of  the  proba- 
bility  curve   must  be  adopted  as  an  indication  of  the  precision 
of  the  results  obtained.     The  function  which  is  commonly  adopted 
as  the  gage  of  precision  is  called  the  probable  error. 

170.  Fundamental  Meaning  of  the  Probable  Error.    By  the 
probable  error  of  a  quantity  is  meant  an  error  of  such  a  magnitude 
that  errors  of  either  greater  or  lesser  numerical  value  are  equally 
likely  to  occur  under  the  same  circumstances  of  observation. 
Or,  in  other  words,  in  any  extended  series  of  observations  the 
probability  is  that  the  number  of  errors  numerically  greater  than 
the  probable  error  will  equal  the  number  of  errors  numerically 
less  than  the  probable  error.     The  probable  error  of  a  single 
observation  thus  becomes  the  critical  value  that  the  numerical 
error  of  any  single  observation  is  equally  likely  to  exceed  or  fall 
short  of.     Similarly  the  probable  error  of  the  arithmetic  mean 

297 


298  GEODETIC  SURVEYING 

becomes  the  critical  value  that  the  numerical  error  of  any  iden- 
tically obtained  arithmetic  mean  is  equally  likely  to  exceed  or 
fall  short  of.  Thus  if  the  probable  error  of  any  angular  measure- 
ment is  said  to  be  five  seconds,  the  meaning  is  that  the  probability 
of  the  error  lying  between  the  limits  of  minus  five  seconds  and  plus 
five  seconds  equals  the  probability  of  its  lying  outside  of  these 
limits.  The  probable  error  is  always  written  after  a  measured 
quantity  with  the  plus  and  minus  sign.  Thus  if  an  angular 
measurement  is  written 

72°  10'  15".8  ±  1".3, 

it  indicates  that  1".3  is  the  probable  error  of  the  given  determina- 
tion. The  probable  error  of  a  quantity  can  not  be  a  positive 
quantity  only,  or  a  negative  quantity  only,  but  always  requires 
both  signs.  It  is  important  to  note  that  the  probable  error  is  an 
altogether  different  thing  from  the  most  probable  error.  Since 
errors  of  decreasing  magnitude  occur  with  increasing  frequency, 
the  most  probable  error  in  any  case  is  always  zero. 

171.  Graphical  Representation  of  the  Probable  Error.  The 
probability  that  an  error  will  fall  between  any  two  given  limits 
(Art.  147)  is  equal  to  the  area  included  between  the  corresponding 
ordinates  of  the  probability  curve.  The  probability  that  an  error 
will  fall  outside  of  any  two  given  limits  must  hence  be  equal  to 
the  sum  of  the  areas  outside  of  these  limits.  If  these  two  proba- 
bilities are  equal,  therefore,  each  such  probability  must  be 
represented  by  one-half  of  the  total  area.  The  probable  error 
thus  becomes  that  error  (plus  and  minus)  whose  two  ordinates 
include  one-half  the  area  of  the  probability  curve.  Referring 
to  Fig.  67,  the  solid  curve  corresponds  to  a  series  of  observations 
taken  under  a  certain  set  of  conditions,  and  the  dotted  curve 
to  a  series  of  observations  taken  under  more  favorable  conditions. 
The  ordinates  yi,  y\,  correspond  to  the  probable  error  r\  of  an 
observation  of  unit  weight  taken  under  the  conditions  pro- 
ducing the  solid  probability  curve,  and  include  between  them- 
selves one-half  of  the  area  of  that  curve.  The  ordinates  y' ',  y', 
correspond  to  the  probable  error  rr  of  an  observation  of  unit 
weight  taken  under  the  conditions  producing  the  dotted  proba- 
bility curve,  and  include  between  themselves  one-half  of  the 
area  of  the  dotted  curve.  The  area  for  any  probability  curve 
(Art.  150)  being  always  equal  to  unity,  it  follows  that  y\,  yi, 


PROBABLE  ERRORS  OF  OBSERVED  QUANTITIES      299 

and  y',  y',  include  equal  areas.  Hence  as  the  center  ordinate  at 
A  grows  higher  and  higher  with  increasing  accuracy  of  observation, 
so  also  must  the  ordinates  y\,  yi,  draw  closer  together.  It  is 
thus  seen  that  the  probable  error  r\  grows  smaller  and  smaller 
as  the  accuracy  of  the  work  increases,  and  therefore  furnishes  a 
satisfactory  gage  of  precision. 


FIG.  67.— The  Probable  Error. 

172.  General  Value  of  the  Probable  Error.  The  area  of  any 
probability  curve  (Art.  150)  equals  unity.  The  area  between 
any  probable  error  ordinates  yi,  yi  (Art.  171),  is  equal  to  half 
the  area  of  the  corresponding  probability  curve.  But  the  area 
between  the  ordinates  yi,  yi  (Art.  147),  is  equal  to  the  probability 
that  an  error  will  fall  between  the  values  x  =  —  ri  and  x  =  +  r\. 
Hence  from  Eq.  (16)  we  have 


h 


(52) 


Since  (Art.  150)  the  precision  of  any  set  of  observations  depends 
entirely  on  the  value  of  ft,  it  follows  that  the  probable  error  ri 
must  be  some  function  of  h.  The  last  member  of  Eq.  (52)  is  not 
directly  integrable,  so  that  the  numerical  relation  of  the  quan- 
tities h  and  n  can  only  be  found  by  an  indirect  method  of  suc- 
cessive approximation  which  is  beyond  the  scope  of  this  volume. 
As  the  r/esult  of  such  a  discussion  we  have, 


0.4769363 
h 


(53) 


It  is  thus  seen  that  for  different  grades  of  work  the  probable  error 
n  varies  inversely  as  the  precision  factor  h. 


300  GEODETIC  SURVEYING 

By  more  or  less  similar  processes  of  reasoning  it  is  also  estab- 
lished that  the  probable  error  of  any  quantity  or  observation 
varies  inversely  as  the  square  root  of  its  weight.  Thus  if  r\  is 
the  probable  error  of  an  observation  of  unit  weight,  then  for  the 
probable  error  rp  of  any  value  with  the  weight  p,  we  have 


(54) 


173.  Direct  Observations  of  Equal  Weight.     From  Eq.  (20) 
we  have 


h  -          -- 
~  \  2St*  ' 

Substituting  this  value  of  h  in  Eq.  (53)    and  reducinjg,  we  have 

•     •  n=  0.6745^1,      •-,••:.•-   '     (55) 

in  which  r\  is  the  probable  error  of  a  single  observation  in  the 
case  of  direct  observations  of  equal  weight  on  a  single  unknown 
quantity,  and  n  is  the  number  of  observations. 
**   Since  in  this  case  (Art.  134)  the   weight   of   the  arithmetic 
mean  is  equal  to  the  number  of  observations,  we  have  (Art.  172), 


-y=  =  0.6745  A  /  ^V  (56) 

\n(n  — 


.          A 
n  n(n  —  1) 


in  which  ra  is  the  probable  error  of  the  arithmetic  mean  in  the 
case  of  direct  observations  of  equal  weight  on  a  single  unknown 
quantity,  and  n  is  the  number  of  observations. 

Example.    Direct  observations  on  an  angle  A  : 

Observed  values                                 v  vz 

29°  21'  59".l  -  2".l  4.41 

29    22    06  .4  +5  .2  27.04 

29    21    58  .1  -3.1  _  9.61 

3)88    06    03  .6  2w2  =  41.06 

2  =  29    22   01  .2  n  =  3 

The  probable  error  of  a  single  observation  is  therefore 


0.6745--  =  0.6745 


PROBABLE  ERRORS  OF  OBSERVED  QUANTITIES      301 

and  of  the  arithmetic  mean,  ** 


o 

V  n      V  3 
whence  we  have 

Most  probable  value  of  A  =  29°  22'  01".2  ±  1".76. 

174.  Direct  Observations  of  Unequal  Weight.     From  Eq.  (2l) 
we  have 


Substituting  this  value  of  h  in  Eq.  (53)  and  reducing,  we  have 

...:/.     .     .     .     (57) 


in  which  n  is  the  probable  error  of  an  observation  of  unit  weight 
in  the  case  of  direct  observations  of  unequal  weight  on  a  single 
unknown  quantity,  and  n  is  the  number  of  observations.  The 
value  of  n  thus  becomes  purely  a  standard  of  reference,  and  it  is 
entirely  immaterial  whether  or  not  any  one  of  the  observations 
has  been  assigned  a  unit  weight.  Having  found  the  value  of 
n  we  have,  from  Eq.  (54), 


in  which  rp  is  the  probable  error  of  any  observation  whose  weight 
is  p. 

Since  in  the  case  of  weighted  observations  (Art.  134)  the  weight 
of  tjhe  weighted  arithmetic  mean  is  equal  to  the  sum  of  the  indi- 
vidual weights,  we  have  (Art.  172), 


^=77^'  0.6745 Jv-fr-iy     .     .     .     (58) 


in  which  rpa  is  the  probable  error  of  the  weighted  arithmetic  mean 
in  the  case  of  direct  observations  of  unequal  weight  on  a  single 
unknown  quantity. 


302  GEODETIC  SURVEYING 

Example.    Direct  base-line  measurements  of  unequal  weight: 

Observed  values        p  pM  v  vz  prfl 

4863.241  ft.       2    9726.482      0.020    0.000400       0.000800 
4863.182  ft.       1    4863.182  -  0.039     0.001521       0.001521 
Sp  =  3)14589.664  Spy2  =  0.002321 

2  =  4863.221  ft.  n  =  2. 

The  probable  error  of  an  observation  of  unit  weight  is  therefore 


0.6745  =  Q.6745X    °-°02321  =  ±  0.032  ft.; 

—  1 


of  an  observation  of  the  weight  2, 


r2  =  _       =  =±  0.023  ft, 

V  p       V  2 

and  of  the  weighted  arithmetic  mean, 

rpa  =  —~  =  ~=  =  ±  0.019ft.; 
Vzp      V  3 

whence  we  have 

Most  probable  value  =  4861.221  ±  0.019  ft. 

175.  Indirect  Observations  on  Independent  Quantities.     From 
Eq.  (22)  we  have 


Substituting  this  value  of  h  in  Eq.  (53)  and  reducing,  we  have 

......     (59) 


in  which  r\  is  the  probable  error  of  an  observation  of  unit  weight 
in  the  case  of  indirect  observations  on  independent  quantities 
(that  is  with  no  conditional  equations)  ,  n  is  the  number  of  observa- 
tion equations,  and  q  is  the  number  of  unknown  quantities. 
Having  found  the  value  of  n,  we  have,  from  Art.  172, 

r\  r\  r\ 

rp  =  —  =,     rx  =  —=,     ry  =  —=,  etc., 
Vp  Vpx  VPy 

in  which  rp  is  the  probable  error  of  any  observation  whose  weight 
is  p,  and  rx  is  the  probable  error  of  any  unknown,  x,  in  terms  of 
its  weight  px,  and  so  on. 


PROBABLE  ERRORS  OF  OBSERVED  QUANTITIES      303 

The  weights  px,  py,  etc.,  of*the  unknown  quantities  are  found 
from  the  normal  equations  by  means  of  the  following 

RULE:  In  solving  the  normal  equations  preserve  the  absolute 
terms  in  literal  form',  then  the  weight  of  any  unknown  quantity  is 
contained  in  the  expression  for  that  quantity,  and  is  the  reciprocal 
of  the  coefficient  of  the  absolute  term  which  belonged  to  the  normal 
equation  for  that  unknown  quanti  y. 

In  applying  the  above  rule  no  change  whatever  is  to  be  made 
in  the  original  form  of  any  normal  equation  until  the  absolute 
term  has  been  replaced  by  a  literal  term.  If  the  normal  equations 
are  correctly  solved  the  coefficients  in  the  literal  expressions  for 
the  unknown  quantities  will  follow  the  same  law  (Art.  162)  as 
the  coefficients  of  normal  equations,  and  this  check  must  never 
be  neglected. 

Example.     Given  the  following  observation  equations  to  determine  the 
most  probable  values  and  the  probable  errors  of  the  unknown  quantities: 

x  +    y  =  10.90  (weights); 

•  2x  —    y  =     1.61  (weight  1); 

x  +  3y  =  24.49  (weight  2). 

Forming  the  normal  equations,  we  have 

9z  +    7y  =    84.90  =  Nx  =  normal  equation  in  x; 
7x  +  22y  =  178.03  =  Ny  =  normal  equation  in  y; 
whence 

x  =       -tfsNx  -  TTTrATj,  =  4.172,  nearly; 
V  =  ~  rhNx  +  T&N,  =  6.765,  nearly; 
and,  by  the  rule, 

Weight  of  x  =  W  =    6.773,  nearly  =  px; 
y  =  if*  =  16.556     "         =py. 

Substituting  in  the  original  equations  the  values  obtained  for  x  and  y,  there 
results 

x  +    y  =  10.937; 
2x  -    y  =    1.579; 

x  +  3y  =  24.467; 

whence,  for  the  residuals,  we  have, 

v,  =  10.90  -  10.937  =  -  0.037  (weight  3); 
vz  =  1.61  -  1.579  =  +0.031  (weight  1); 
v3  =  24.49  -  24.467  =  +  0.023  (weight  2). 

We  therefore  have  for  the  probable  error  of  ail  observation  of  unit  weight, 


A/  '^^  I 


=  0.6745A  =  ±  0.053; 


304  GEODETIC  SURVEYING 

for  the  probable  error  of  x, 

r,  0.053 

rx  =  —  =  -     —  =  =  ±  0.020; 
VP        V6.773 


rv  =  -~=  =     /  =  ±  0.013; 


and  for  the  probable  error  of  y, 

/ 

V16.556 
whence  we  write 

x  =  4.172  ±  0.020  and      y  =  6.765  ±  0.013. 

176.  Indirect  Observations  Involving  Conditional  Equations. 
From  Eq.  (23)  we  have 


Substituting  this  value  of  h  in  Eq.  (53)  and  reducing,  we  have 

(60) 


in  which  r\  is  the  probable  error  of  an  observation  of  unit  weight 
in  the  case  of  indirect  observations  involving  conditional  equa- 
tions, n  is  the  number  of  observation  equations,  q  is  the  number 
of  unknown  quantities,  and  c  is  the  number  of  conditional  equa- 
tions. Having  found  the  value  of  n,  we  have,  from  Art.  172, 

rp  =  -^L,     rx  =  -y=,     rv  =  -~=,     etc., 
Vp  Vpx  Vpy 

in  which,  as  in  the  previous  article,  rp  is  the  probable  error  of  any 
observation  whose  weight  is  p,  and  rx  is  the  probable  error  of  any 
unknown,  x,  in  terms  of  its  weight  px,  and  so  on. 

In  order  to  find  the  value  of  the  weights  px,  py,  etc.,  the  con- 
ditional equations  are  first  eliminated  (Art.  166),  and  the  normal 
equations  due  to  the  resulting  observation  equations  are  then 
treated  by  the  rule  of  the  preceding  article.  By  repeating  the 
process  with  different  sets  of  unknowns  eliminated,  the  weight 
of  each  unknown  will  eventually  be  determined. 

177.  Other  Measures  of  Precision.  The  measures  of  precision 
thus  far  introduced  are  the  precision  factor  h,  and  the  probable 
error  r.  Two  other  measures  of  precision  are  sometimes  used, 


PROBABLE  ERRORS  OF  OBSERVED  QUANTITIES       305 

and  are  of  great  theoretic  vakie.  These  are  known  as  the  mean 
error,  and  the  mean  absolute  error. 

By  the  mean  error  is  meant  the  square  root  of  the  arithmetic 
mean  of  the  squares  of  the  true  errors. 

By  the  mean  absolute  error  (often  called  the  mean  of  the  errors) 
is  meant  the  arithmetic  mean  of  the  absolute  values  (numerical 
values)  of  the  true  errors. 

Referring  to  Fig.  68,  the  precision  factor  h  is  equal  to  \?n 
times  the  central  ordinate  AY.  Considering  either  half  of  the 


Point  of  Inflection 


\.    Point  of  Inflection 


FIG.  68. — Measures  of  Precision. 


curve  alone,  the  ordinate  for  the  probable  error  r  bisects  the 
included  area,  the  ordinate  for  the  mean  absolute  error  T)  passes 
through  the  center  of  gravity,  and  the  ordinate  for  the  mean 
error  s  passes  through  the  center  of  gyration  about  the  axis  A  Y. 
The  ordinate  for  e  also  passes  through  the  point  of  inflection 
of  the  curve. 

The  measure  of  precision  most  commonly  used  in  practice  is 
the  probable  error  r,  but  as  the  different  measures  bear  fixed 
relations  to  each  other  a  knowledge  of  any  one  of  them  determines 
the  value  of  all  the  others,  as  shown  in  the  following  summary: 


Precision  factor         =  h. 
Probable  error  =  r  = 


0.4769363 


Mean  absolute  error  =  y  =  — — -  =  1.1829r. 


Mean  error 


=  — !— =  1.4826  r. 


306  GEODETIC  SUKVEYING 


B.  OF  COMPUTED  QUANTITIES 

178.  Typical    Cases.     When    the   probable    error    is  known 
for  each  of  the  quantities  from  which  a  computed  quantity  is 
derived,  the  probable  error  of  the  computed  quantity  may  also 
be  determined.     Any  problem  which  may  arise  will  come  under 
one  or  more  of  the  five  following  cases : 

1.  The  computed  quantity  is  the  sum  or  difference  of  an  observed 
quantity  and  a  constant. 

2.  The  computed  quantity  is  obtained  from  an  observed  quantity 
by  the  use  of  a  constant  factor. 

3.  The  computed  quantity  is  any  function  of  a  single  observed 
quantity. 

4.  The  computed  quantity  is  the  algebraic  sum  of  two  or  more 
independently  observed  quantities. 

5.  The  computed  quantity  is  any  function  of  two  or  more  inde- 
pendently observed  quantities. 

The  fifth  case  is  general,  and  embraces  all  the  other  ca  es. 
The  first  four  cases,  however,  are  of  such  frequent  occurrence  that 
special  rules  are  developed  for  them.  Any  combination  of  the  rules 
is  therefore  admissible  that  does  not  violate  their  fundamental 
conditions,  since  the  first  four  rules  are  only  special  cases  of  the 
fifth  rule. 

179.  The  Computed  Quantity  is  the  Sum  or  Difference  of  an 
Observed  Quantity  and  a  Constant. 

Let  u  and  ru  =  the  computed  quantity  and  its  probable  error; 
x  and  rx  =  the  observed  quantity  and  its  probable  error; 
a  =  a  constant; 

then 

u  =  ±  x  ±  a; 
and 

ru  =  rx (61) 

It  is  evidently  immaterial  whether  x  is  directly  observed  or 
is  the  result  of  computation  on  one  or  more  observed  quantities. 
The  only  essential  condition  is  satisfied  if  rx  is  the  probable  error 
of  x.  If  £  is  a  computed  quantity  the  probable  error  rx  may  be 
derived  by  any  one  of  the  present  rules. 


PROBABLE  ERRORS  OF  COMPUTED  QUANTITIES      307 

Example.    Referring  to  Fig.  69,  4>he  most  probable  value  of  the  angle  x  is 
x  =  30°  45'  17".22  =t  1".63. 

What  is  the  most  probable  value  of  its  supplement  y,  and  the  probable  error 
of  this  value  ? 

From  the  conditions  of  the  problem  we 
have 

y  =  180°  -  x; 
whence 


ru  =  rx  =  db  1".63,  FIG.  69. 

and 

y  =  149°  14'  42".78  ±  1".63. 

180.  The  Computed  Quantity  is  Obtained  from  an  Observed 
Quantity  by  the  Use  of  a  Constant  Factor. 

Let  u  and  ru  =  the  computed  quantity  and  its  probable  error; 
x  and  rx  =  the  observed  quantity  and  its  probable  error; 
a  =  a  constant; 

then 

u  =  ax 
and 

ru  =  arx.      .  .  ;    .:    .     .     .     .     (62) 

Evidently,  as  in  the  previous  case,  x  may  be  any  function  of  one 
or  more  observed  quantities,  provided  that  rx  is  its  correct  probable 
error.  The  rule  of  this  article  is  only  true  when  the  constant  a 
represents  a  strictly  mathematical  relation,  such  as  the  relation 
between  the  diameter  and  the  circumference  of  a  circle.  Staking 
out  100  feet  by  marking  off  successively  this  number  of  single 
feet  is  not  such  a  case,  as  the  total  space  staked  out  is  not  neces- 
sarily exactly  100  times  any  one  of  the  single  spaces  as  actually 
marked  off.  In  all  probability  some  of  the  feet  will  be  too  long 
and  others  will  be  too  short,  so  that  (owing  to  this  compensating 
effect)  the  total  error  will  be  very  much  less  than  100  times  any 
single  error,  and  the  probable  error  must  be  found  by  Art.  182. 
In  the  case  of  the  circle,  however,  the  circumference  is  of  neces- 
sity exactly  equal  in  every  case  to  TT  times  the  diameter. 


308  GEODETIC  SURVEYING 

Example.  The  radius  of  a  circle,  as  measured,  equals  271.16  ±  0.04  ft. 
What  is  the  most  probable  value  of  the  circumference,  and  the  probable 
error  of  this  value? 

Circumference  =  271.16  X  2x  =  1703.75  ft.; 
ru=  rxX  2x  =  ±  0.04  X  2x  =  ±  0.25  ft.; 

whence  we  write 

Circumference  =  1703.75  d=  0.25  ft. 

181.  The  Computed  Quantity  is  any  Function  of  a  Single 
Observed  Quantity. 

Let  u  and  ru  =  the  computed  quantity  and  its  probable  error  ; 
x  and  rx  =  the  observed  quantity  and  its  probable  error; 
then 

u  =  #(x); 
and 


Evidently,  as  in  the  two  previous  cases,  x  may  be  any  function 
of  one  or  more  observed  quantities,  provided  that  rx  is  its  correct 
probable  error. 

Example.     The  radius  j  of  a  circle  equals  42.27  d=  0.02  ft.     What  is 
the  most  probable  value  and  the  probable  error  of  the  area? 

u  =  xx2  =  (42.27) 2  X  x  =  5613.26; 
du  =  Ziacdx,  —  =  2xz, 

ru  =  TS  ^*  =  ra;(2xz)  =  ±  0.02  X  2x  X  42.27  =  ±  5.31; 

whence  we  write 

Area  =  5613.26  d=  5.31  sq.ft. 

182.  The  Computed  Quantity  is  the  Algebraic  Sum  of  Two  or 
More  Independently  Observed  Quantities. 

Let  u  and  ru  =  the    computed    quantity    and    its    probable 

error; 

x,  y,  etc.  =  the  independently  observed  quantities; 
rx)  ry,  etc.  =  the  probable  errors  of  x,  y,  etc.;  j 

then 

u  =  ±  x  ±  y  ±  etc.; 
and 

(64) 


PROBABLE  ERRORS  OF  COMPUTED  QUANTITIES      309 

The  observed  quantities  x,  &  z,  etc.,  may  each  be  a  different 
function  of  one  or  more  observed  quantities,  but  the  absolute 
independence  of  x,  y,  z,  etc.,  must  be  maintained.  In  other 
words,  x  must  be  independent  of  any  observed  quantity  involved 
in  y,  z,  etc.;  y  independent  of  any  observed  quantity  involved 
in  x,  z,  etc. ;  and  so  on.  Thus,  for  instance,  we  can  not  regard 
2x  as  equal  to  x  +  x,  and  substitute  in  the  above  formula,  since 
x  and  x  in  the  quantity  2x  are  not  independent  quantities. 
Attention  is  also  called  to  the  fact  that  the  signs  under  the 
radical  are  always  positive,  whether  the  computed  quantity  is 
the  result  of  addition  or  subtraction  or  both  combined. 

Example  1.     Referring  to  Fig.  70,  given 

x  =  70°  13'  27".60  ±  2".16; 
y  =  40    57    19  .32  ±  1  .07; 

to  find  the  most  probable  value  and  the  probable  error  of  z. 
In  this  case 

z  =  x  +  y  =  111°  10'  46".92; 


ru  =  V(2.16)2  +  (1.07) 2  =  ±2 

whence  we  write 

z  =  111°  10'  46".92  ±2".41. 


FIG.  70.  FIG.  71. 

Example  2.     Referring  to  Fig.  71,  given 

x  =  70°  13'  27".60  ±  2".16; 
y  =  40    57    19  .32  ±  1  .07; 

to  find  the  most  probable  value  and  the  probable  error  of  z. 
In  this  case 

z  =  x  -  y  =  29°  16'  08".28; 

ru  =  V(2.16)2  +  (1.07)2  =  ±  2".41; 
whence  we  write 

z  =  29°  16'  08".28  ±  2".41. 


310  GEODETIC  SURVEYING 

183.  The  Computed  Quantity  is  any  Function  of  Two  or  More 
Independently  Observed  Quantities. 

Let  u  and  ru  =  the  computed  quantity  and  its  probable  error; 

x}  y,  etc.  =  the  independently  observed  quantities; 
rx,  ry,  etc.  =  the  probable  errors  of  x,  y,  etc.  ; 
then 

u  =  <£(z,  y,  etc.); 
and 


du\2 


,_ 

•  • 


All  the  remarks  under  the  previous  case  apply  with  equal  force 
to  the  present  case. 

Example  1.     The  measured  values  for  the  two  sides  of  a  rectangle  are 

x  =  55.28  ±  0.03  ft. 
y  =  85.72  ±  0.05  ft. 

What  is  the  most  probable  value  of  the  area  and  its  probable  error? 

u  =  Xy  =  55.28  X  85.72  =  4738.60; 
du  du 


=  V(0.03  X  85.72) 2  +  (0.05  X  55.28) 2  =  ±  3.78; 
whence  we  write 

Area  =  4738.60  ±  3.78  sq.ft.  «. 

Example  2.      Referring  to  the  right-angled 
triangle  in  Fig.  72,  given 

x  =  38.17  ±  0.05  ft.;  FIG.  72. 

y  =  19.16  ±0.04  ft.; 

to  find  the  most  probable  value  of  the  hypothenuse  u  and  its  probable  error. 


u  =  Vx2  +  y2  =  V(38.17)2  +  (19.16)2  =  42.71; 
du  x  du  y 


dx       Vx*         2  dy       V 


//       rrx     \2        I      rvy      \2  =      l(rxxY  + 

/(38.17  X  0.05)2  +  (19.16  X  0.04)2  _ 
=  \  "          (38.17)2  +  (19.16)2 


PROBABLE  ERRORS  OF  COMPUTED  QUANTITIES      311 

whence  we  write  & 

Hypothenuse  =  42.71  ±  0.05  ft. 

Example  3.     Referring  to  Fig.  73,  in  which  the   horizontal  distance  x 
and  the  vertical  angle  </>  have  been  measured, 
given 

x  =  489.11  ±  0.32  ft.; 
$  =  12°  17'±  1'; 


FIG.  73. 

to  find  the  most  probable  value  of  the  elevation  u  and  its  probable  error 
u  =  x  tan  <f>  =  106.49; 

du  •    .                       du           x 
—  =  tan  <j),  J2  = 2T~' 


cos 


It  is  necessary  at  this  point  to  remember  that  expressing  an  angle  in  degrees, 
minutes,  and  seconds,  is  only  a  trigonometrical  convenience,  and  that  the 
true  measure  of  an  angle  is  the  ratio  of  the  subtending  arc  to  its  radius. 
An  arc  expressed  in  minutes  must  therefore  be  compared  with  a  radian  ex- 
pressed in  minutes  (that  is,  an  arc  whose  length  equals  that  of  the  describing 
radius)  in  order  to  complete  its  angular  meaning. 


1  radian  =  3438',  nearly. 


3438 


(0.32  tan  <;6)2  +  (-    -X 


whence  we  write 


489.] 
cos2 


u  =  106.49  ±  0.16  ft. 


CHAPTER  XIV 
APPLICATION  TO  ANGULAR  MEASUREMENTS 

184.  General  Considerations.  In  the  adjustment  of  angular 
measurements  three  classes  of  problems  may  arise,  known  as 
single  angle  adjustment,  station  adjustment,  and  figure  adjust- 
ment. 

By  single  angle  adjustment  is  meant  the  determination  of  the 
most  probable  value  of  an  angle  which  can  be  obtained  from  the 
measurements  made  directly  upon  it. 

By  station  adjustment  is  meant  the  determination  of  the  most 
probable  values  of  two  or  more  angles  at  a  single  station,  in  order 
to  meet  the  condition  of  being  geometrically  consistent. 

By  figure  adjustment  is  meant  the  determination  of  the  most 
probable  values  of  the  angles  involved  in  any  geometric  figure, 
in  order  to  meet  the  condition  of  being  geometrically  consistent. 

In  trigonometric  work  of  any  importance  each  individual 
angle  is  always  measured  a  large  number  of  times,  and  the  most 
probable  value  due  to  these  results  is  considered  as  its  measured 
value.  The  station  adjustment  or  figure  adjustment  is  then 
made  in  accordance  with  the  conditions  of  the  given  case. 

SINGLE  ANGLE  ADJUSTMENT 

185.  The  Case  of  Equal  Weights.  In  this  case  (Art.  155) 
the  most  probable  value  is  the  arithmetic  mean  of  the  individual 
measurements. 

Example.  Three  equally  reliable  measurements  of  the  angle  x  give 
29°  21'  59".l,  29°  22'  06".4,  29°  21'  58".  1.  What  is  its  most  probable 
value? 

29°  21'  59".l 
29  22  06  .4 
29  21  58  .1 
3)88  06  03  .6 
29°  22'  01".2 

The  most  probable  value  is  therefore  29°  22'  01".2. 

312 


APPLICATION  TO  ANGULAR  MEASUREMENTS 


313 


186.  The  Case  of  Unequal JWeights.  ln  this  case  (Art.  157) 
the  most  probable  value  is  the  weighted  arithmetic  mean  of  the 
individual  measurements. 

Example.  Three  measurements  of  an  angle  x  give  38°  15'  17". 2  (weight  1), 
38°  15'  15".5  (weight  3),  and  38°  15'  18".0  (weight  2).  What  is  its  most 
probable  value? 


38°  15'  17".2  X  1 
38  15  15  .5  X  3 
38  15  18  .0  X  2 


38°  15'  17".2 
114    45    46  .5 
76    30    36  .0 
6)229    31  "39~7~ 
38°  15'  16".6 

The  most  probable  value  is  therefore  38°  15'  16".6. 

STATION  ADJUSTMENT 

187.  General   Considerations.     All   cases  of  station   adjust- 
ment necessarily  imply  one  or  more  conditional  equations.     In 
the  determination  of  the  most  probable  values  of  the  several 
angles  these  equations  may  be  avoided  (Art.  165),  eliminated 
(Art.  166),  or  involved  in  the  computa- 
tion (Art.  167),  as  found  most  convenient. 

The  angles  at  a  station  are  in  general 
measured  under  similar  conditions,  so 
that  in  making  the  adjustment  it  is 
customary  to  give  to  each  angle  a  weight 
equal  to  the  number  of  observations  (or 
the  sum  of  the  weights  in  the  case  of 
weighted  observations)  on  which  it  de- 
pends. Angles  are  seldom  measured  a 
sufficient  number  of  times  to  make  it 
justifiable  to  weight  them  inversely  as 

the  squares  of  their  probable  errors,  as  would  be  required  by 
the  last  paragraph  of  Art.  172.  The  following  cases  of  station 
adjustment  show  the  general  principles  involved: 

188.  Closing   the    Horizon   with   Angles    of   Equal   Weight. 
Referring  to  Fig.  74, 

Let  x,  y,  Zj  .  .  .  w  =  the  angles  measured; 
a,b,c,...m  =  their  measured  values; 
n  =  the  number  of  angles  measured; 
d  =  (a -\-b-\-c. ..-}-m)—  360°  =  the  discrepancy  to 
be  adjusted; 


FIG.  74. 


r*3lTYCF  CALIFORNIA 
£NT  OF  CIVIC  ENGINES 


314  GEODETIC  SURVEYING 

then  the  observation  equations  are 

x  =  a; 

z  =  c] 

w  =  m; 
and  the  conditional  equation  is 

x  +  y  +  z  .  .  .  +w  =  360°. 

It  is  evident  £rom  the  figure,  however,  that  this  conditional 
equation  may  be  avoided  (Art.  165)  by  regarding  all  the  angles 
except  w,  for  instance,  as  independent,  and  involving  the  required 
condition  by  expressing  this  angle  in  terms  of  the  others.  The 
observation  equations  thus  become 

x  =  a; 

V  =  b'> 
z  =  c] 

360°  -  (x+  y  +  z  .  .  .)  =  m. 

Passing  to  the  reduced  observation  equations  (Art.  163)  by  sub- 
stituting for  the  most  probable  values  of  the  unknown  quantities, 

x  =  a  +  vi't 
y  =  b  +  v2', 
z  =  c  +  v3; 

etc. 
we  have 

vi  =  0; 
v2  =  0; 
^3  =  0; 

vi  +  v2  +  vz  .  .  .  =  360°  -  (a  +  b  +  c  .  .  .  +  m)  =  -  d; 
giving  the  normal  equations 

2^i  -|-    i>2  ~h    #3  ~h  t>4  -f-  V&  .  .  .  =  —  d\ 
v\  -\~  2z^2  H~    v3  ~\~  DA  ~\~  V&  •  •  •  =  —  d] 

etc.,  etc. 


APPLICATION  TO  ANGULAR  MEASUREMENTS        315 

Subtracting  the  second  equation  from  the  first,  we  have 

vi  —  V2  =  0,     or     vi  =  V2- 
Subtracting  the  third  equation  from  the  second,  we  have 

^2   —   #3    =   0,       Or       V2    =   V^. 

Or,  in  general, 

vi  =  V2  =  v%  =  v4  =  v^  —  etc. 
But 

Vl  +  V2  +  V3    •   •   •   =    —  d' 

whence 

vi  =  v2  =  i'3  =  etc.  =  -   — .      ....     (66) 

Eq.  (66)  shows  that  when  angles  of  equal  weight  are  arranged 
around  a  point  so  as  to  close  the  horizon,  the  most  probable  value 
for  each  angle  is  found  by  a  uniform  dis- 
tribution of  the  discrepancy. 

Example.     Referring    to    Fig.    75,    the    following 
observations  are  to  be  adjusted: 

x  =  45°  20'  19".3  (weight  1); 
y  =  151  52  48  .6  (weight  1); 
z  =  162  46  58  .4  (weight  1). 

360°     00'     06".3 
360      00      00  .0 


d  =  +  06".3  FIG.  75. 

In  accordance  with  the  above  principle  each  angle  must  be  reduced  by  2".l, 
giving  for  the  most  probable  values 

x  =  45°  20'  17".2; 
y  =  151  52  46  .5; 
z  =  162  46  56  .3. 

189.  Closing  the  Horizon  with  Angles  of  Unequal  Weight. 
Referring  to  Fig.  74,  page  313, 

Let   x,  y,  z,  .  .  .w  =  the  angles  measured; 

a,  b,  c,  .  .  .  m  =  their  measured  values; 
Pit  P2,  Ps>  •  •  •  Pn  =  their  respective  weights; 

n  =  the  number  of  angles  measured ; 
d=(a  +  6  +  c...+w)-  360°  =  the  discrepancy  to 
be  adjusted; 


316  GEODETIC  SURVEYING 

then  the  observation  equations  are 

x  =  a  (weight  p\) ; 
y  =  b  (weight  p2); 
z  =  c  (weight  ps) ; 

w  =  m  (weight  pn)', 
and  the  conditional  equation  is 

x  +  y  +  z  .  .  .  +  w  =  360°. 

It  is  evident  that  this  conditional  equation  may  be  avoided,  as  in 
Art.  188,  by  writing  the  observation  equations  in  the  form 

x  =  a  (weight  pi) ; 
y  =  b  (weight  p2) ; 
z  =  c  (weight  ps); 

360°  -  (x  +  y  +  z  .  .  .)  =  m  (weight  pa). 

Passing  to  the  reduced  observation  equations,   as  before,   by 
substituting 

x  =  a  +  vi', 

y  =  b  +  V2] 
etc.; 

we  have 

vi  =  0  (weight  pi) ; 
v2  =  0  (weight  p2) ; 
v3  =  0  (weight  p3); 

vi  +  v2  +  v-3  .  .  .  =  -  d     (weight  pj ; 
giving  the  normal  equations 

+  PH(VI  +  V2  +  v3  .  .  .)  =  -  pnd; 

+  pn(Vl   +  V2  +  ^3  •   •   •)   =     -  Pnd', 
+  pn(Vi   +  V2  +  V3  .   .   .)   =    -  pnd. 

etc.,  etc. 


APPLICATION  TO  ANGULAE  MEASUREMENTS        317 

Subtracting  the  second  equation  from  the  first,  the  third  equation 
from  the  second,  and  so  on,  we  have 


-  P2V2  =  0,     or     pivi  =  p2V2', 

P2V2  —  P3V3   =  0,       Or       p2V2   =  P^V^ 

etc.,  etc.; 

or,  in  general, 

=  etc.      .     .     .     (67) 


Eq.(67)  shows  that  when  angles  of  unequal  weight  are  arranged 
around  a  point  so  as  to  close  the  horizon,  the  most  probable  value 
for  each  angle  is  found  by  distributing  the  discrepancy  inversely 
as  the  corresponding  weights. 

Example.     Referring  to  Fig.  75,  page  315,  the  following  observations  are 
to  be  adjusted  : 

x=  50°  49'  27".6  (weight  2); 
y  =  149  22  22  .8  (weight  1); 
z  =  159  48  05  .9  (weight  3). 

359°     59'     56".3 
360      00      00  .0 


d  =  -  03".7 

In  accordance  with  the  above  principle  this  discrepancy  is  to  be  distributed 
as 

111. 
2     :     1     :     3' 

which,  cleared  of  fractions,  equals 

3:6:2. 

The  three  corrections  are  thus 

3.7  X  A  =  1".01,       3.7  X  TT  =  2".02,       and      3.7  X  TT  =  0".67. 
The  most  probable  values  are  therefore 

x  =  50°  49'  28".61; 
y  =  149  22  24  .82; 
z  =  159  48  06  .57. 

190.  Simple  Summation  Adjustments.     Referring  to  Fig.  76, 
page  318,  let  x,  y,  z,  etc.,  represent  a  series  of  angles  at  the  point  C, 


318 


GEODETIC  SURVEYING 


and  let  w  represent  the  corresponding  summation  angle, 
we  must  have  geometrically, 

w  =  x  +     -\-z  +  etc. 


Then 


But  the  measured  values  of  these  angles  will  seldom  satisfy  this 
conditional  equation,  and  an  adjustment  becomes  necessary  to 

remove  the  discrepancy.  In  making  the 
adjustment  it  is  evidently  immaterial 
whether  we  regard  w  or  wf  as  the  angle 
actually  measured,  since  these  values 
are  mutually  convertible  and  only  differ- 
ent expressions  for  the  same  fundamental 
idea.  The  adjustment  may  therefore  be 
made  in  any  case  by  subtracting  the 
measured  value  of  w  from  360°  to  obtain 
the  apparent  value  of  w' ',  and  then 
applying  the  rule  of  Arts.  188  or  189, 
as  may  be  necessary.  Since  the  correc- 
tion to  w'  will  have  the  same  sign  as  all  the  remaining  corrections, 
it  is  evident  that  the  correction  to  w  must  have  the  opposite 
sign.  We  are  thus  led  to  the  following  conclusions : 

In  the  case  of  equal  weights  the  most  probable  values  of  the 
measured  angles  are  obtained  by  an  equal  numerical  distribu- 
tion of  the  discrepancy,  with  opposite  signs  for  the  summation- 
angle  correction  and  all  the  remaining  corrections. 

In  the  case  of  unequal  weights  the  most  probable  values  of  the 
measured  angles  are  obtained  by  a  numerical  distribution  of  the 
discrepancy  inversely  proportional  to  the  several  weights,  with 
opposite  signs  for  the  summation-angle  correction  and  all  the 
remaining  corrections. 


Example  1. 
adjusted: 


Referring  to  Fig.  77,  the  following  observations  are  to  be 


x  =  39°  12'  32".6  (weight  1); 
y  =  44  47  59  .3  (weight  1); 
y  =  84  00  35  .8  (weight  1). 

39°  12'  32".6 

44  47  59  .3 

84  00  31  .9 

84  00  35  .8 

3)03_.9 

1  .3 


FIG.  77. 


APPLICATION  TO  ANGULAE  MEASUREMENTS        319 

In  accordance  with  the  above  principles  the  most  probable  corrections  to 
the  measured  angles  are    <- 


giving  as  the  most  probable  values, 

x  =  39°  12'  33".9; 

y  =  44    48    00  .6; 

x  +  y  =  84    00    34  .5. 

Example  2.  Referring  to  Fig.  77,  the  following  observations  are  to  be 
adjusted: 

x  =  40°  16'  23".7  (weight  2); 

y  =  46    36    48  .5  (weights); 

x  +  y  =  86     53    08  .0  (weight  4). 

40°  16'  23".7 
46  36  48  .5 
86  53  12  .2 
86  53  08  .0 
d  =  04  .2 

In  accordance  with  the  above  principles  this  discrepancy  is  to  be  distributed 
numerically  as 

1  I         L. 

2  3          4  ' 

which,  cleared  of  fractions,  equals 

6:4:3; 
giving  as  the  most  probable  corrections 

-  4.2  X  A  =  -  1".94; 
-4.2  XA  =  -  I". 29; 
+  4.2  X  A  =  +  0".97; 

and  therefore  as  the  most  probable  values 

x  =  40°  16'  21".76; 

y  =  46    36    47    21; 

x  +  y  =  86    53    08  .97. 

191.  The  General  Case.  The  cases  given  in  Arts.  188,  189,  and 
190,  are  the  only  ones  in  which  it  is  desirable  to  establish  special 
rules.  Any  case  of  station  adjustment  may  be  solved  by  writ- 
ing out  the  observation  and  conditional  equations  and  then  apply- 
ing the  principles  developed  in  Chapters  XI  and  XII. 


320 


GEODETIC  SURVEYING 


Example  1.     Referring  to  Fig.  78,  find  the  most  probable  values  of  the 
angles  x,  y,  and  z,  from  the  following  observations: 

x  =  25°  17'  10".2  (weight  1); 

y  =  28    22  16  .4  (weight  2); 

z  =  32    40  28  .5  (weight  2); 

x  +  y  =  53    39  23  .1  (weight  2); 

x  +  y  +  z  =  86     19  57  .8  (weight  1). 

Letting  y,,  v2,  v3,  be   the  most  probable  corrections  for  x,  y,  and    z,  we  may 
write  (Art.  163)  the  reduced  observation  equations 

v,                    =  0".0  (weight  1); 

v2           =  0  .0  (weight  2) ; 

v3  =  0.0  (weight  2) ; 

Vl  +  v2           =  -  3  .5  (weight  2);  . 

•      »i  +  v2  +  *>3  =  +  2  -7  (weight  1); 


FIG.  78. 
from  which  we  have  the  normal  equations 


FIG.  79. 


4vi  +  3v2  +    i>3  =  -  4.3; 

3wi  +  5v2  +    y3  =  -  4.3; 

fi  +    *>2  +  3v3  =  2.7; 
whose  solution  gives 

vi  =  -  1".04,     v2  =  -  0".52,      v3  =  +  1".42. 
The  most  probable  values  of  the  given  angles  are  therefore 

x  =  25°  17'  09".16; 
y  =  28  22  15  .88; 
z  =  32  40  29  .92. 

Example  2.     Referring  to  Fig.  79,  find  the  most  probable  values  of  the 
angles  x,  y,  and  z,  from  the  following  observations: 

x  =    14°  11'  17".  I  (weight  1); 

y  =  19  07  21  .3  (weight  2); 

x  +  y  =  33  18  43  .4  (weight  1); 

z  =  326  41  18  .2  (weight  2); 

y  +  z  =  345  48  39  .2  (weight  3). 


APPLICATION  TO  ANGULAR  MEASUREMENTS        321 

As  the  angles  x,  y,  and  z  close  the  h$r izon  they  must  satisfy  the  conditional 
equation 

x  +  y  +  z  =  360°. 

Avoiding  this  conditional  equation  by  subtracting  all  angles  containing 
z  from  360°,  we  have 

x  =  14°  11'  17".l  (weight  1); 

y  =  19    07    21  .3  (weight  2); 

x  +  y  =  33     18    43  .4  (weight  1); 

x  +  y  =  33     18    41   .8  (weight  2); 

x  =  U     11    20  .8  (weight3); 

in  which  x  and  y  may  be  regarded  as  independent  quantities. 

Letting  v^  and  v2  be  the  most  probable  corrections  for  x  and  y,  and 
writing  the  reduced  observation  equations  in  accordance  with  Art.  163, 
we  have 

Vl  =  0".0  (weight  1); 

v2  =  0  .0  (weight  2); 
Vi  +  v2  =  5  .0  (weight  1); 
fi  +^2  =  3  .4  (weight  2); 
vi  =  3  .7  (weight3); 

from  which  we  have  the  normal  equations 

7vi  +  3v2  =  22.9; 
3wi  +  5v2  =  11.8; 
whose  solution  gives 

»i  =  +  3".04,     v2  =  +  0".53. 

The  most  probable  values  of  x  and  y  are  therefore 

a  =    14°  11'  20".14; 
y  =    19    07    21  .83; 

and  hence  the  most  probable  value  for  z  must  be 
z  =  326°  41'  18".03, 
in  order  to  make  the  sum  total  of  360°. 

FIGURE  ADJUSTMENT 

192.  General  Considerations.  All  cases  of  figure  adjust- 
ment necessarily  imply  one  or  more  conditional  equations.  In 
the  determination  of  the  most  probable  values  of  the  several 
angles  these  equations  may  be  avoided  (Art.  165),  eliminated 
(Art.  166),  or  involved  in  the  computation  (Art.  167),  as  found 
most  convenient.  The  angles  in  a  triangulation  system  are  in 
general  measured  under  similar  conditions,  so  that  in  making  the 
adjustment  it  is  customary  to  give  to  each  angle  a  weight  equal  to 
the  number  of  observations  (or  the  sum  of  the  weights  in  the  case 
of  weighted  observations)  on  which  it  depends.  Angles  are  sel- 
dom measured  a  sufficient  number  of  times  to  make  it  justifiable 
to  weight  them  inversely  as  the  squares  of  their  probable  errors, 


322  GEODETIC  SURVEYING 

as  would  be  required  by  the  last  paragraph  of  Art.  172.  In  work  of 
moderate  extent  any  required  station  adjustment  may  be  made 
prior  to  the  figure  adjustment,  but  in  very  important  work  it  may 
be  desirable  to  make  both  adjustments  in  one  operation.  Except 
in  very  important  work,  the  triangles,  quadrilaterals,  or  other 
figures  in  a  system  may  be  adjusted  independently.  In  work  of 
the  highest  importance  the  whole  system  would  be  adjusted  in 
one  operation.  The  following  cases  of  figure  adjustment  show 
the  general  principles  involved,  assuming  that  the  reduction  for 
spherical  excess  (Arts.  56,  57,  58)  has  already  been  made. 

193.  Triangle   Adjustment   with   Angles    of   Equal   Weight. 
Referring  to  Fig.  80, 


FIG.  80. 

Let  x,  y,  z  =  the  unknown  angles; 
a,b,c  =  the  measured  values; 

d  =  (a  +  b  +  c)  —  180°  =  the    discrepancy    to   be 
adjusted. 

Avoiding  the  conditional  equation  (Art.  163)  for  the  sum  of  the 
three  angles  by  writing  the  observation  equations  in  terms  of 
x  and  y  as  independent  quantities,  we  have 

x  =  a; 
y  =  b; 
x  +  y  =  180°  -  c. 

Substituting  for  the  most  probable  values 

x  =  a  +  vi; 
y  =  b  +  V2; 
we  have 

vi  =  0; 

v2  =  0; 
Vl  +  V2  =  180°  -  (a  -f  b  +  c)  =  -  d; 


APPLICATION  TO  ANGULAR  MEASUREMENTS         323 

giving  the  normal  equations,    & 

•2vi  +    V2  =  -  d] 
vi  +  2v2  =  -  d; 
whence  by  subtraction, 

vi  —  V2  =  0,     or     vi  =  V2. 

In  a  similar  manner  it  may  be  shown  that  v\  or  V2  is  equal  to 
v3,  or  in  general, 

Vl   =  V2  =  V%. 

Bat  evidently, 

vi  +  v2  +  v3  =  -  d] 
whence, 

Vl  =  v2  =  vs  =  -  g (68) 

Equation  (68)  shows  that  when  the  measured  angles  of  a  tri- 
angle are  considered  of  equal  weight,  the  most  probable  values  of 
these  angles  are  found  by  adjusting  each  angle  equally  for  one-third 
of  the  discrepancy. 

Example.     The  measured  values  (of  equal  weight)  for  the  three  angles 

of  a  triangle  are  92°  33'  15".4,  48°  11'  29".6,  and  39°  15'  12". 3.     What  are 
the  most  probable  values? 

Measured  Values  Most  Probable  Values 

92°  33'  15".4  92°  33'  16".3 

48    11    29  .6  48    11    30  .5 

39    15    12  .3  39     15    13  .2 

179°  59'  57".3  180°  00'  00".0 
180    00    00  .0 
3)  -  02".7 


194.  Triangle  Adjustment  with  Angles  of  Unequal  Weight. 

Referring  to  Fig.  80, 

Let  Xj  y,  z  =  the  unknown  angles; 
a,  b,  c  =  the  measured  values; 
PI,  P2,  PS  =  the  respective  weights; 

d  =  (a  +  b  +  c)  —  180°  =  the  discrepancy  to  be 

adjusted. 

Avoiding  the  conditional  equation  as  before  by  making  x  and  y 
the  independent  quantities,  we  have 

x  =  a  (weight  pi); 

y  =  b  (weight  p2); 

x  +  y  =  180°  -  c    (weight  p3). 


324  GEODETIC  SURVEYING 

Substituting,  as  before, 

x  =  a  +  vi; 
y  =  b  +  V2', 
we  have 

vi  =  0  (weight  pi); 

V2  =  0  (weight  p2); 

vi  +  v2  =  180°  -(a  +  b  +  c)  =  -d     (weight  p8); 

giving  the  normal  equations 


whence,  by  subtraction, 

—  P2V2  =  0,     or    pivi  =  p2V2. 


In  a  similar  manner  it  may  be  shown  that  p\v\  or  p2V2  is  equal  to 
p3fl3.     Hence,  in  any  case, 


Eqs.  (69)  show  that  when  the  measured  angles  of  a  triangle  are 
considered  of  unequal  weight,  the  most  probable  values  of  these 
angles  are  found  by  distributing  the  discrepancy  inversely  as  the 
corresponding  weights. 

Example.  The  measured  values  for  the  three  angles  of  a  triangle  are 
97°  49'  56".8  (weight  2),  38°  06'  05".0  (weight  1),  and  44°  04'  01".l  (weight  3). 
What  are  the  most  probable  values? 

97°  49'  56".8 
38  06  05  .0 
44  04  01  .1 


180°  00'  02".9 

180    00    00  .0 

d  =  +  02".9 

\:  |  :  |  =  3:6:2;        3+6  +  2  =  11; 

+  02.9  X  A  =  +  00".79,         +  02.9  X  TT  =  +  01".58, 

+  02.9  X  A  =  +  00".53. 
The  most  probable  values  are  therefore 

97°  49'  56".01 

38    06    03  .42 

44    04    00  .57 

180°  00'  00".00 


APPLICATION  TO  ANGULAR  MEASUREMENTS        325 


195.  Two  Connected  Triangles.  A  simple  case  of  figure 
adjustment  is  illustrated  in  Fig.  81.  Two  triangles  are  here 
connected  by  the  common  side  AB,  and  the  eight  indicated 
angles  are  measured.  It  is  evident  from  the  figure  that  four 
independent  conditional  equations  must  be  satisfied  by  the 
adjusted  values  of  the  angles,  for  the  summation  angles  at  A  and  B 
must  agree  with  their  component  angles,  and  the  angles  in  each 
of  the  two  triangles  must  add  up  to  180°.  The  problem  may  be 
worked  out  by  the  methods  of  Arts.  165,  166,  or  167.  The  fol- 


FIG.  81.— Two  Connected  Triangles. 

lowing  example  is  worked  out  by  the  algebraic  elimination  of  the 
conditional  equations  (Art.  166)  in  order  to  illustrate  this  method. 

Hr 

Example.     Referring  to  Fig.  81,  given  the  following  observed  values  of 
equal  weight,  to  find  the  most  probable  values  of  the  measured  angles: 


B3 


=  65C 
=  75 
=  47 
=  53 


Observed  Values  of  Angles 
25'     18".l;  A  =  141' 


43 
26 
19 


45  .1; 
11  .9; 

51  .8; 


B  = 
C  = 
D  = 


100 
67 
50 


09' 
46 
08 
56 


02".2; 
06  .6; 
28  .4; 
25  .2. 


Writing  out  the  four  conditional  equations,  we  have 

A  =  Ai  +  A2', 

B  =  B3+Bt', 

C  +  A!  +  B,  =  180°; 

D  +  A2  +  Bi  =  180°. 

In  accordance  with  Art.  166,  four  of  the  unknowns  may  be  regarded  as  inde- 
pendent quantities.  It  is  evident  from  the  figure  or  from  the  conditional 
equations  that  Ai,  A 2,  Bz,  B\  may  be  taken  as  the  independent  unknowns, 
whence  A,  B,  C,  D  become  the  dependent  ones.  These  latter  quanti- 
ties are  selected  as  the  dependent  ones  in  preference  to  any  other 


326  GEODETIC  SURVEYING 

four  because  they  are  so  easily  found  from  the  given  conditional  equations. 
Solving  for  these  quantities,  we  have 

A  =  Ai  +  A2; 

B  =  Jj3  -\~  Bi'f 

C  =  180°  -  (Ai  +  B3)', 

D  =  180°  -  (At  +  B,). 

Substituting  in  the  observation  equations  and  reducing,  we  have 

Ai  =  65°  25'  18".l;  Al  +  A2  =  141°  09'  02".2; 

A2  =  75  43  45  .1;  B3  +  B4  =  100  46  06  .6; 

B3  =  47  26  11  .9;  Al  +  B,  =  112  51  31  .6; 

B,  =  53  19  51   .8;  A*  +  B,  =  129  03  34  .8. 

Letting  »i,  v2,  vs,  v4,  be  the  most  probable  corrections  for  A\,  A2,  B3,  J54, 
respectively,  we  may  write  the  reduced  observation  equations  (Art.  163) 
as  follows: 

vi  =  0".0;  Vi  +  v2  =  -  1".0; 

V2  =  o  .0;  v3  +  Vi  =  +  2  .9; 

v3  =  0  .0;  vi  +  v3  =  +  1  .6; 

v4  =  0  .0;  vi  +  vt  =  -  2  .1. 

In  a  simple  case  like  this  the  reduced  observation  equations  would  usually 
be  written  directly  from  the  figure  instead  of  going  through  the  above  alge- 
braic work.  Having  decided  on  the  proper  independent  quantities,  these 
equations  are  simply  written  so  as  to  represent  the  apparent  discrepancy 
in  each  observation,  always  subtracting  the  independent  quantities  from 
the  values  they  are  compared  with.  Forming  the  normal  equations,  we  have 

o..       i_      ..       I  i     r\//  f* . 

»i  +3v2  +    v4  =  -  3  .1; 

^i  +  3v3  +    ^4  =  -f  4  .5; 

v2  +    v3  +  3v4  =  +  0  .8; 
whose  solution  gives 

t>i  =  +  0".10,  v3  =  +  1".41, 

vt=  -1  .13,  vt  =  +  0  .17. 

Using  these  corrections  to  find  A\,  A%,  B3,  and  B^  and  then  the  conditional 
equations  to  find  A,  B,  C,  and  D,  we  have  for  the  most  probable  values 

A!  =  65°  25'  18".20;  A  =  141°  09'  02".17; 

A2  =  754  43  43  .97;  B  =  100  46  05  .28; 

B3  =  47  26  13  .31;  C  =    67  08  28  .49; 

£4  =  53  19  51  .97;  D  =    50  56  24  .06. 

196.  Quadrilateral  Adjustment.  The  best  method  to  use  in 
adjusting  a  geodetic  quadrilateral,  Fig.  82,  is  the  method  of 
correlatives,  Art.  167.  In  accordance  with  Art.  58  the  adjusted 
angles  must  satisfy  the  following  three  angle  equations : 

.     (70) 


APPLICATION  TO  ANGULAE  MEASUREMENTS        327 


and  also  the  following  side  equation: 

sin  a  sin  c  sin  e  sin  g 


t, 


sin  6  sin  d  sin  /  sin  h 
which  may  be  written  in  the  logarithmic  form 

"S  log  sin(a,  c,  e,g)  -  2  log  sin (6,  d,  f,  h) 


0. 


(71) 


FIG.  82.  —  The  Geodetic  Quadrilateral. 

Letting  Ma,  Mb,  etc.,  represent  the  measured  values  of  the 
angles  a,  6,  etc.,  and  l\,  li,  Z3,  l±,  represent  the  discrepancies  in 
these  equations  due  to  the  errors  in  the  measured  angles,  we  have 

S(Mato  ^0- 


(Mc  +  Md)-(Mg+Mh)  =  l3 

2  log  sin  (Ma,  Mc,  Me,  Mg)  -2  log  sin  (Mb,  Md,  Mf, 

The  corrections  va,  vb,  etc.,  to  be  added  algebraically  to  the 
measured  values  Ma,  Mb,  etc.,  must  reduce  these  equations  to 
zero  in  order  that  the  conditional  equations  (70)  and  (71)  may  be 
satisfied.  Therefore  we  must  have 

Va+      Vb  +      Vc  + 
Va+      Vb 


\    -   (73 


Ve~      Vf  =   - 

Vc  +       Vd  Vg—       Vh  =   - 

dava  —  dbvb  +  dcvc  —  ddvd  f  de  ve  -  dfvf  -f-  dgva  -  dh  rh  =  — 

in  which  va,  vbj  etc.,  are  to  be  expressed  in  seconds,  and  in  which 
da,  db)  etc.,  are  the  tabular  differences  for  one  second  for  the 


328 


GEODETIC  SURVEYING 


log  sin  Ma)  log  sin  Mb,  etc.  If  any  angle  is  greater  than  90°  it 
is  evident  that  the  corresponding  tabular  difference  must  be 
considered  negative,  since  the  sine  will  then  decrease  as  the  angle 
increases  in  value.  The  conditional  Eqs.  (73)  being  in  the  form 
of  Eqs.  (43),  the  most  probable  values  of  va,  vb,  etc.,  may  now 
be  found  by  the  method  of  correlatives  (Art.  167),  by  means  of 
Eqs.  (49)  and  (50)  .  Re-writing  these  equations  with  the  symbols 
used  in  the  present  article,  and  remembering  that  there  are  four 
conditional  equations  and  hence  four  correlatives  required,  we 
have  in  the  general  case,  from  Eqs.  (49)  and  (73), 


p 


TH  2^p~l 

,oc     ^   2—4- 
'  P       2      P 


+  k2Z- 


, 
--  f- 

p 


p 


]—  =  -  zs 

p 

;-  =  -/. 
jp        3 


.     (74) 


and  from  Eqs.  (50)  and  (73), 

va  =  ki h  k2— 

Pa  Pa 

J_  j_ 


'<   =    V. 


tV  ==  fci-- 


^i 

p^ 


Pf 
^4   4S 

3-  -  A4^ 

PA  Ph 


.     (75) 


APPLICATION  TO  ANGULAR  MEASUREMENTS        329 


in  which  pa  represents  the  weigbt  of  Ma,  pb  the  weight  of  Mb,  and 
so  on. 

In  the  case  of  equal  weights  we  have,  from  Eqs.  (73)  and  (74), 

(db  +  dd+df+ddya  =  -h 

(da  —db  -de  +  df)k±  =  —  12 

=  -Z3     (76) 


and  from  Eqs.  (75), 


=  ki  +  k2  -  dbk± 


vf 


b  = 

Mb 

+   Vbm, 

f  = 

Mf  - 

f  v/; 

c  = 

Me 

-MeS 

g  = 

Mg    -{ 

-      Vg' 

d  = 

Md 

+  vd', 

h  = 

Mh- 

h  vh. 

.     .     .     (77) 


i  —  k'2  —  dfk± 
Vv  =  ki  —  k3  +  dgk± 
Vh  =  hi  ~  k3  —  dhk4 

Having  found  the  values  of  va,  vb,  etc.,  we  have  in  any  case  for 
the  most  probable  values  of  the  angles  a,  b,  etc., 

a  =  M  a  +  va',  e  =  Me  +  ve] 


(78) 


197.  Other  Cases  of  Figure  Adjustment.  There  is  evidently 
no  limit  to  the  number  of  cases  of  figure  adjustment  that  may  be 
made  the  subject  of  consideration,  but  few  of  them  are  likely  to 
be  of  interest  to  the  civil  engineer.  Any  case  that  may  arise  may 
be  adjusted  by  the  method  of  correlatives  (Art.  167),  similarly  to 
the  quadrilateral  adjustment  (Art.  196),  provided  the  observa- 
tion equations  and  conditional  equations  are  properly  expressed. 
In  any  case  the  conditional  equations  must  cover  all  the  geo- 
metrical conditions  which  must  be  satisfied,  and  at  the  same  time 
must  be  absolutely  independent  of  each  other.  The  number  of 


330 


GEODETIC  SURVEYING 


ill 


I     i 
I  a  I  r= 


I      I 


8 


o 

i-H 
2  2       I 


5! 


I    r 


I  -^ 


5    8 


l! 


s 


^  I 


a 


± 


53 


s 


w 


00    00    (M    o 
CO*    CO    t>J    t-I 

CO 

CO 


+  1 


I  + 


o 

I  I 


o  o 

I  I 

II  II 


OS    O    <M 

9  d  o 


I     I 


co     oo 

V. 
CO 


g 


APPLICATION  TO  ANGULAR  MEASUREMENTS        331 


independent  conditional  equations  can  always  be  ascertained  by 
subtracting  the  number  of  independent  quantities  from  the  number 
of  observed  quantities.  The  number  of  independent  quantities 
is  in  general  easily  determined  by  an  inspection  of  the  given  figure, 
being  that  number  of  independent  values  which  fixes  a  single 
location  for  each  angular  point.  A  study  of  the  following  exam- 
pies  will  illustrate  the  principles  involved. 

Example  1.  Referring  to  Fig.  83,  the  base  A  B  and  the  indicated  angles 
have  been  measured;  determine  the  number  and  nature  of  the  independent 
conditional  equations. 

It  is  evident  from  the  figure  that  it  will  take  two  angles  from  the  fixed 
points  A  and  B  to  locate  either  C  or  D,  and  that  these  four  angles  are 
independent.  We  may  therefore  select  Ai,  At,  B^  B%,  as  independent 
angles,  and  as  this  will  fix  the  points  C  and  D  it  will  also  fix  the  values  of 
the  angles  Cl  and  C2,  so  that  we  can  not  have  more  than  four  independent 
angles.  In  this  particular  case  any  four  of  the  angles  can  be  taken  as  the 
independent  ones,  but  this  freedom  of  choice  is  not  a  general  rule.  As  there 
are  six  observations  of  which  only  four  are  independent,  it  follows  (Art.  166) 
that  two  independent  conditional  equations  must  be  involved.  Starting 
from  any  known  side,  A  B,  we  may  in  general  compute  any  other  line  of  a 
system  through  two  different  sets  of  triangles,  and  the  requirement  that  these 
two  results  shall  be  identical  will  always  lead  to  a  corresponding  side  equa- 
tion. In  the  present  case,  therefore,  the  two  conditional  equations  must 
consist  of  one  angle  equation  and  one  side  equation.  The  angle  equation  is 
evidently, 

A!  +  A2  +  Bl  +  B2  +  Ci  +  C2  =  180°. 

Taking  C  D  as  a  convenient  line  from  which  to  determine  the  side  equation, 
and  equating  its  values  as  computed  through  the  triangles  A  B  D  and  A  C  D, 
and  through  the  triangles  A  B  D  and  BCD,  the  side  equation  is  easily 
found  to  be, 

sin  AI  sin  BI  sin  d  =  sin  A2  sin  B2  sin  C2. 


FIG.  83. 


FIG.  84. 


Example  2.  Referring  to  Fig.  84,  the  base  A  B  and  the  indicated  angles 
have  been  measured;  determine  the  number  and  nature  of  the  independent 
conditional  equations. 

In  this  case,  as  in  the  previous  one,  four  independent  angles  will  fix  the 
whole  figure,  so  that  the  fact  that  nine  angles  have  been  measured  demands 
the  existence  of  five  independent  conditional  equations,  as  nine  minus  four 


332  GEODETIC  SURVEYING 

equals  five.  In  regarding  any  four  of  the  angles  as  independent,  it  is  evident 
that  no  three  of  them  must  lie  in  any  one  triangle,  as  this  would  at  once 
destroy  the  independence  of  these  three  angles  by  setting  a  condition  on 
their  sum.  Since,  as  explained  in  Example  1,  there  must  be  one  side  equa- 
tion, on  account  of  the  one  known  line  A  B,  it  follows  that  the  present  case 
must  involve  four  independent  angle  equations  to  make  up  the  total  of 
five  independent  conditional  equations  required.  An  examination  of  the 
figure,  however,  furnishes  five  angle  equations,  as  follows: 

Ai  +  C2  +  D2  =  180° 

A2  +  Bl  +  D,  =  180° 

#2  +  Ci  +  Di  =  180° 

Al  +  A2  +  #1  +  #2  +  Ci  +  C2  =  180° 

A  +  D2  +  D,  =  360° 

As  there  can  be  but  four  independent  angle  equations,  it  follows  that 
any  one  of  these  five  must  be  dependent  on  the  other  four.  An  examination 
of  the  equations  will  show  at  once  that  any  one  of  them  may  be  derived 
from  the  remaining  four.  We  may  therefore  choose  any  four  of  these  five 
equations  for  our  four  angle  equations.  Since  the  figure  is  identical  with 
the  one  in  Example  1,  our  side  equation  as  before  will  be, 

sin  Ai  sin  BI  sin  C\  =  sin  A%  sin  B%  sin  Cz. 

Example  3.  Referring  to  Fig.  85,  the  base  A  B  and  the  indicated  angles 
have  been  measured,  the  interior  station  being  a  random  point  not  purposely 

falling  on  any  diagonal  of  the  figure; 
determine  the  number  and  nature  of  the 
independent  conditional  equations. 

In  this  case  the  angles  in  any  five  of 
the  six  triangles  will  fix  the  whole  figure; 
and  since  there  can  be  but  two  indepen- 
dent angles  in  each  of  the  five  triangles 
so  selected,  it  follows  that  we  must  have 
ten  independent  angles.  As  there  are 
eighteen  measured  angles  and  ten  inde- 
pendent angles,  we  must  have  eight  inde- 
pendent conditional  equations.  As  before 
there  must  be  one  side  equation,  leaving 
seven  angle  equations  required.  Eight  such 
-  85.  equations  may  be  formed,  to  meet  the  con- 

ditions that  six  triangles  must  each  contain 

180°,  that  the  corner  angles  of  the  hexagon  must  add  up  to  720°,  and  that 
the  central  angles  must  add  up  to  360°.  Any  seven  of  these  eight  angle 
equations  may  be  taken  as  the  independent  ones,  when  the  requirement  of 
the  other  one  will  also  be  satisfied.  For  the  side  equation  we  may  com- 
pute any  side,  such  as  E  D,  by  going  around  the  figure  in  both  directions 
from  AB,  from  which  it  will  appear,  as  in  the  previous  examples,  that 
the  product  of  the  sines  of  one  set  of  alternate  corner  angles  must  equal 
the  product  of  the  sines  of  the  other  set  of  alternate  corner  angles, 


CHAPTER  XV 

APPLICATION  TO  BASE-LINE  WORK 

198.  Unweighted  Measurements.  If  a  base  line  is  measured 
from  end  to  end  a  number  of  times  in  the  same  manner,  and 
under  such  conditions  that  the  different  determinations  of  its 
length  may  be  regarded  as  of  equal  weight,  then  (Art.  155)  the 
arithmetic  mean  of  the  several  results  is  the  most  probable  value 
of  its  length.  The  probable  error  of  a  single  measurement 
(Art.  173)  is  given  by  the  formula 


n  =  0.6745  J-^-r,      .......     (79) 

\/  Tl>          J_ 

and  the  probable  error  of  the  arithmetic  mean  (Art.  173)  of  n 
measurements  by  the  formula 


>•<,  =  -;=  =    0.6745  ^re(re_1} (SO) 

Example.     Direct  base-line  measurements  of  equal  weight: 

Observed  Values  v  v2 

6717 . 601  ft.  -  0 . 025  0 . 000625 

6717 . 632  ft.  +  0 . 006  0 . 000036 

6717.645  ft.  +  0.019  0.000361 

3)20152. 878  ft.'  2v2  =  0.001022 

z=    6717.626ft.  n  =  3 

ri  =  0.6745  J2»_±  0.0152  ft. 


0.0088  ft. 


V3 

Most  probable  value  =  6717.626  ±  0.0088  ft. 

199.  Weighted  Measurements.     If  a  base  line  is    measured 
from  end  to  end  a  number  of  times  in  the  same  manner,  but  under 

333 


334 


GEODETIC  SURVEYING 


such  conditions  that  the  different  determinations  of  its  length 
must  be  regarded  as  of  unequal  weight,  then  (Art.  157)  the  weighted 
arithmetic  mean  of  the  several  results  is  the  most  probable  value 
of  its  length.  The  probable  error  of  single  measurement  of 
unit  weight  (Art.  174)  is  given  by  the  formula 


(81) 


=  0.6745  .  /-  •— 


the  probable  error  of  any  measurement  of  the  weight  p  (Art.  174) 
by  the  formula 


rp  =  — L  =  0.6745  A  /  ^PV\.  ,  (82) 

V/>  \p(n  - 

and  the  probable  error  of  the  weighted  arithmetic  mean  (Art.  174) 
by  the  formula 


--—=    =--  °-6745  J^^^Tfy        •      •      (& 


(S3) 


Example.     Direct  base-line  measurements  of  unequal  weight : 


Observed  Values 

p 

pM 

7829.614ft. 

1 

7829.614 

-C 

7829.657ft. 

2 

15659.314 

+  c 

7829.668ft. 

1 

7829.668 

+  c 

7829.628ft. 

3 

23488.884 

-C 

Sp 

=  7 

)54807.480 

* 

z  =7829.  640  ft. 

1A7«    /0.002470 

0.000676  0.000676 
0.000289  0.000578 
0.000784  0.000784 
0 . 000144  0.000432 
Ziw2  =  0.002470 


n  =  4 


=  =fc  0.0194  ft. 


V2 


=  ±  0.0137  ft. 


V3 


=  ±  0.0112  ft. 


rpfl=—        =  ±0.0073  ft. 

V7 

Most  probable  value  =  7829.640  ±  0.0073  ft. 

200.  Duplicate  Lines.  In  work  of  ordinary  importance  or 
moderate  extent  it  is  sufficient  to  measure  a  base  line  twice  and 
average  the  results  for  the  adopted  length.  When  the  same  line 


APPLICATION  TO  BASE-LINE  WORK  335 

is  measured  twice  with  equal  carg  it  is  called  a  duplicate  line.  The 
rules  of  Art.  198  necessarily  include  duplicate  lines,  but  this 
case  is  of  such  frequent  occurrence  that  special  rules  are  found 
convenient  for  the  probable  errors.  Letting  d  represent  the  dis- 
crepancy between  the  two  measurements,  and  remembering  that 
the  arithmetic  mean  is  the  most  probable  value,  we  have 

d  d 

vi  =  +  g     and     ^2  =   -  2  • 

Substituting  these  values  in  Eq.  (79)  and  replacing  n  with  r\  for 
the  case  of  duplicate  lines,  we  have  for  the  probable  error  of  a 
single  measurement  of  the  length  I, 

ri  =  0.4769Vd2  =  ±  o.4769d.     .     .     .     (84) 

Substituting  the  same  values  in  Eq.  (80),  we  have  for  the  probable 
error  of  the  arithmetic  mean, 

ra  =  ±  0.3373  d; (85) 

whence 

ra  (approximately)  =  ±  \d (86) 

Example.     Measurement  of  a  duplicate  base  line: 

Observed  Values 

4998.693  ft.  0.4769  X  0.034  =  0.0162. 

4998.659  ft.  0.3373  X  0.034  =  0.0115. 

d  =  0.034  ft. 

r,  =  =t  0.0162  ft.  ra  =  ±  0.0115  ft. 

Most  probable  value  =  4998.676  ±  0.0115  ft. 

201.  Sectional  Lines.  A  base  line  may  be  divided  up  into 
two  or  more  sections,  and  each  section  measured  a  number  of 
times  as  a  separate  line.  Each  section,  on  account  of  its  several 
measurements,  will  thus  have  a  most  probable  length  and  a  prob- 
able error  independent  of  any  other  section  of  the  line.  If 
h,  h,  •  •  .  ^,be  the  most  probable  lengths  of  the  several  sections, 
then  (Art.  168)  the  most  probable  length  L  for  the  whole  line,  is 

L  =  h  +  k  ...  +  /„  =  SZ.       .  ".    .    .     (87) 

And  if  n,  r2,  .  .  .  rn,  be  the  probable  errors  of  the  several  values 
Zi,  12,  etc.,  then  (Art.  182)  the  probable  error  rL  for  the  whole 

line,  is 

rL  =  vV  +  r22  .  .  .  +  Tr?  =  V2r2.    .     .   ..     (88) 


336  GEODETIC  SURVEYING 

Example.     Sectional  base-line  measurement.     Given 

Zi  =  3816.172  ±  0.022  ft. 
h  =  4122.804  =fc  0.019  ft. 
13  =  3641.763  ±  0.017  ft. 

L  =  3816.172  +  4122.804  +  3641.763  =  11580.739  ft. 


TL  =  V  (0.022) 2  +  (0.019)2  +  (0.017)2  =  ±  0.034  ft. 
Most  probable  value  L  =  11580.739  ±  0.034  ft. 

202.  General  Law  of  the  Probable  Errors.     In  measuring  a 
base  line  bar  by  bar  or  tape-length  by  tape-length,   the  case  is 
essentially  one  of  sectional  measurement  (Art.  201),  in  which 
each  section  is  measured  a  single  time,  and  in  which  each  full 
section  is  of  the  same  measured  bar-  or  tape-length.     If  the  con- 
ditions remain  unchanged  throughout  the  measurement,  therefore, 
the  probable  error  will  be  the  same  for  each  full  section.     As 
explained  in  Art.  180,  however,  this  is  not  a  case  of  computed 
values  depending  on  a  constant  factor,  so  that  the  probable  error 
of  the  whole  line  will  not  follow  the  law  of  that  article. 
Let  L  =  the  total  length  for  a  line  of  full  sections; 
rL  =  the  probable  error  of  this  line ; 
t  =  the  length  of  the  measuring  instrument; 
rt  =  the  probable  error  for  each  length  measured; 
n  =  the  number  of  lengths  measured; 

then  (Art.  201) 

rL 
But  evidently 


whence 

(89) 


Eq.  (89)  is  derived  on  the  assumption  that  only  full  bar-  or  tape- 
lengths  are  used.  The  fractional  lengths  that  occur  at  the  ends 
of  a  base  (or  elsewhere)  form  such  a  small  proportion  of  the  total 
length,  however,  that  no  appreciable  error  can  arise  by  assuming 
Eq.  (89)  as  generally  true.  A  consideration  of  the  various 
methods  and  instruments  used  in  measuring  base  lines  also  shows 


APPLICATION  TO  BASE-LINE  WORK  337 

that  in  any  case  nothing  but  systematic  errors  could  modify  the 
truth  of  this  equation.  -  We  may  therefore  write  as  a 

GENERAL  LAW  :  Under  the  same  conditions  of  measurement  the 
probable  error  of  a  base  line  varies  directly  as  the  square  root  of  its 
length. 

From  the  manner  in  which  this  law  has  been  derived  it  is 
evident  that  it  is  theoretically  true  whether  the  length  assigned 
to 'a  base  line  is  the  result  of  a  single  measurement,  or  the  average 
of  a  number  of  measurements,  so  long  as  the  lines  being  compared 
have  all  been  measured  in  the  same  way.  In  cases  where  the 
given  lines  have  been  measured  more  than  once,  so  that  each 
line  has  its  own  direct  probable  error,  we  can  not  expect  an  exact 
agreement  with  the  law.  But  this  relation  of  the  probable 
errors  is  more  likely  than  any  other  that  can  be  assigned,  and 
hence  shows  the  relative  accuracy  that  may  be  reasonably  expected 
in  lines  of  different  length.  The  chief  point  of  interest  in  the  law 
lies  in  the  fact  that  the  error  in  a  base  line  is  not  likely  to  increase 
any  faster  than  the  square  root  of  its  length,  so  that  the  probable 
error  where  a  line  is  made  four  times  as  long  should  not  be  more 
than  doubled,  and  so  on. 

Example.  A  base  line  measured  under  certain  conditions  has  the  value 
7716.982  ±  0.028  ft.  What  is  the  theoretical  probable  error  of  a  base  line 
15693.284  ft.  long,  measured  under  the  same  conditions? 


Theoretical  probable  error  of  new  line  =  d=  0.0399  ft. 

203.  The  Law  of  Relative  Weight.  In  accordance  with 
the  law  of  the  previous  article,  we  may  write  for  the  probable 
error  of  a  base  line  of  any  length 

rL  =  mVL,    '.     .     .     .     .     .     .     (90) 

in  which  m  is  a  coefficient  depending  on  the  conditions  of  measure- 
ment.    Also  in  accordance  with  the  law  of  Art.  172,  we  may  write 


rL  =  s  —=, 
V 


1 
p 


in  which  p  is  the  weight  assigned  to  the  line  and  s  is  a  coefficient 
depending  on  the  unit  of  weight  and  the  conditions  of  measure- 


338  GEODETIC  SURVEYING 

ment.    Since  the  unit  of  weight  is  entirely  arbitrary  we  may  assign 
that  value  to  p  which  will  make  s  equal  m,  and  write 

r^m^P  ' (91) 

Combining  Eqs.  (90)  and  (91),  we  have 

mVL  =  m— 7= ; 
Vp 

from  which 


whence  we  have  the 

GENERAL  LAW  :  Under  the  same  conditions  of  measurement  the 
weight  of  a  base  line  varies  inversely  as  its  length. 

From  the  manner  in  which  this  law  has  been  derived  it  is 
evident  that  it  is  theoretically  true  whether  the  length  assigned 
to  a  base  line  is  the  result  of  a  single  measurement,  or  the  average 
of  a  number  of  measurements,  provided  the  lines  compared  have 
all  been  measured  in  the  same  way. 

If  two  or  more  base  lines  are  measured  under  different  con- 
ditions, they  may  be  first  weighted  so  as  to  offset  this  circum- 
stance, and  then  weighted  inversely  as  their  lengths.  The 
relative  weight  of  each  line  will  then  be  the  product  of  the  weights 
applied  to  it. 

204.  Probable  Error  of  a  Line  of  Unit  Length.  The  probable 
error  of  an  angular  measurement  conveys  an  absolute  idea  of  its 
precision  without  regard  to  the  size  of  the  angle.  The  pro':  able 
error  of  a  base  line,  however,  conveys  no  idea  of  the  precision 
of  the  work  unless  accompanied  by  the  length  of  the  line.  It  is 
therefore  convenient  to  reduce  the  probable  error  of  a  base  line 
to  its  corresponding  value  for  a  similar  line  of  unit  length.  A 
unit  of  comparison  is  thus  established  for  different  grades  or 
pieces  of  work  which  is  independent  of  the  length  of  the  bases. 
Such  a  unit  has  no  actual  existence,  but  is  purely  a  mathematical 
basis  of  comparison. 

From  Eq.  (89)  we  have 


t 


APPLICATION  TO  BASE-LINE  WORK  339 

Hence,  when  L  equals  1,  we  have  for  r0,  the  probable  error  of  a 
unit  length  of  line, 


whence  in  general 

rL  =  r0\/L,      ....;..     (93) 

in  which  all  the  values  refer  to  single  measurements.  From  this 
equation  we  see  that  the  probable  error  of  any  base  line  is  equal 
to  the  square  root  of  its  length  multiplied  by  the  probable  error 
of  a  unit  length  of  such  a  line.  If  r0  is  well  determined  for 
given  instruments,  conditions,  and  methods,  Eq.  (93)  informs  us 
in  advance  what  is  a  suitable  probable  error  for  a  single  measure- 
ment, and  hence  (Art.  198)  for  the  average  of  any  number  of 
measurements  of  a  line  of  the  given  length  L.  The  base-line 
party  therefore  knows  whether  its  work  is  up  to  standard,  or 
whether  additional  measurements  are  required. 

205.  Determination  of  the  Numerical  Value  of  the  Probable 
Error  of  a  Line  of  Unit  Length.     From  Eq.  (93)  we  have, 


whence 

TL 


(94) 


So  that  in  any  case  where  the  length  of  a  line  and  the  correspond- 
ing probable  error  are  known,  the  formula  determines  a  value 
for  TQ.  In  order  for  the  value  of  TQ  to  be  reliable  it  must  be  based 
on  many  such  determinations,  but  the  expense  prohibits  many 
measurements  of  a  long  base  line.  As  the  law  is  known,  however, 
which  connects  the  values  of  the  probable  error  for  all  lengths 
of  line,  it  is  just  as  satisfactory  to  determine  ro  from  much  shorter 
lines,  which  may  be  quickly  and  cheaply  measured  many  times. 
The  usual  plan  is  to  measure  a  series  of  duplicate  lines,  so  that  the 
probable  error  for  a  single  measurement  is  known  in  each  case 
from  the  discrepancy  in  each  pair  of  lines.  Since  all  results  are 
reduced  to  the  same  unit  length  it  is  immaterial  whether  the 
different  duplicate  lines  are  of  equal  length  or  not. 


340  GEODETIC  SUEVEYING 

In  accordance  with  Eq.  (84)  we  have,  for  any  single  measure- 
ment of  the  duplicate  line  I, 


ri  =    . 
whence,  in  accordance  with  Eq.  (94), 


but,  in  accordance  with  Eq.  (92),  we  have  for  any  length  of  line  I 


whence 

r0  =  0.4769  Vp<F,        .     .     .     .     .     .     .     .     (95) 

when  determined  from  a  single  duplicate  line.  If  a  number  of 
duplicate  lines  are  measured  we  will  have  a  corresponding  number 
of  values  (r0)i,  (7*0)2,  etc.,  based  on  the  discrepancies  di,  c?2,  etc., 
of  the  several  duplicate  lines.  It  might  at  first  be  supposed  that 
the  average  value  of  these  determinations  of  ro  would  best  repre- 
sent the  result  of  all  the  measurements.  What  is  really  wanted, 
however,  is  that  value  of  r0  which  gives  equal  recognition  to  the 
conditions  which  caused  its  different  values.  A  just  recognition 
of  each  value  of  r0,  therefore,  will  require  us  to  consider  equal 
sections  of  any  line  as  having  been  measured  respectively  under 
those  conditions  that  produced  the  several  values  of  r0.  The 
probable  error  for  the  whole  line  is  then  found  from  the  probable 
errors  of  the  different  sections,  and  this  result  reduced  to  the 
probable  error  of  a  unit  length. 

Let  n  =  the  number  of  values  (r0)i,  (r0)2,  etc.; 
L  =  the  length  of  any  given  line; 

whence  the  required  equal  sections  will  be 

(L\         (L\  L 

(-)   =   — )   =  etc-  — r* 

Wi       Wa  n' 

and,  in  accordance  with  Eq.  (93), 

r  (L\=  (r0)i\->      r,/,x  =  (rJtJ-,     etc.; 


APPLICATION  TO  BASE-LINE  WOEK 
I 

whence,  in  accordance  with  Eq.«(88), 


341 


and,  in  accordance  with  Eq.  (94), 


(96) 


but,  in  accordance  with  Eq.  (95), 

(r0)1  =  0.4769v/pdi5,     (r0)2 
so  that 

2r02 
whence 

r0=  0.4769 


(97) 


when  determined  from  a  number  of  duplicate  lines.  In  using 
formulas  (95)  and  (97)  it  is  to  be  remembered  that  d  is  the  dis- 
crepancy in  any  duplicate  line,  p  is  the  weight  (reciprocal  of  the 
length)  of  that  line,  n  is  the  number  of  duplicate  lines,  and  r0 
is  the  probable  error  of  a  single  measurement  of  a  line  of  unit 
length. 

Example.     Determination  and  application  of  the  probable  error  of  a  base 
line  of  unit  length: 


Duplicate  Lines                d 

512011*"    1        0<006 

d2                   p 

0.000036         5T3- 

Pd* 
0.0000000703 

619.184ft.   j             n 
619.176  "    j        °-00 

0.000064         6T9 

0.0000001034 

750  971  f"    1        °-009 

0.000081        yir 

0.0000001079 

619'l84f"    }        °-°04 
750^972  ^   }        °-012 

0.000016         «T¥ 
0.000144        yir 

0.0000000258 
0.0000001917 

from  which  we  have 

2pd2  =  0.0000004991       and      n 
whence 

=  5; 

04-eoV0-0000004"1-  '  ««""«* 

5 

'' 

342  GEODETIC  SURVEYING 

which  is  therefore  the  probable  error  for  a  single  measurement  of  one  foot 
made  under  the  given  conditions.  For  a  single  measurement  of  a  base 
line  of  any  length  L,  therefore,  made  under  these  same  conditions,  the 
probable  error  would  be,  in  accordance  with  Eq.  (93), 

rL  =  r0Vl  =  ±  0.000151  VZ"  ft. 
Thus  if  L  is  10,000  feet,  we  would  have 

rL  =  ±  0.000151  X  VlOOOO  =  ±  0.0151  ft. 

And  if  such  a  line  were  measured  four  times  we  should  have,  theoretically,  for 
the  probable  error  of  the  average  length, 

ra  =  ±  0.0151  -5-  VT  =  ±  0.0076  ft. 

It  thus  becomes  known  in  advance  what  probable  error  is  to  be  expected 
under  the  given  conditions. 

206.  The  Uncertainty  of  a  Base  Line.  By  the  uncertainty 
of  a  base  line  is  meant  the  value  obtained  by  dividing  its  probable 
error  by  its  length.  In  accordance  with  Art.  202,  the  probable 
error  of  a  base  line  varies  as  the  square  root  of  its  length,  so  that 
the  probable  error  increases  much  more  slowly  than  the  length 
of  the  line.  On  account  of  the  greater  opportunity  for  the 
compensation  of  errors,  therefore,  long  lines  are  relatively  more 
accurate  than  short  lines.  While  the  unit  probable  error  r0 
very  satisfactorily  indicates  the  grade  of  accuracy,  whether  a 
line  be  long  or  short,  it  does  not  furnish  any  idea  of  the  degree  of 
accuracy  with  which  the  length  of  a  given  line  is  known.  The 
uncertainty  of  a  base  line,  however,  shows  at  once  the  precision 
attained  in  its  measurement.  If  r\  be  the  probable  error  of  a 
single  measurement  of  a  base  line  whose  length  is  /,  then  for  the 
uncertainty  Ui  of  a  single  measurement,  we  have 


and  for  the  uncertainty  Ua  of  the  arithmetic  mean  of  n  measure- 
ments, 

U  =  -a  =  -H- 
l        iVn' 

But,  in  accordance  with  Eq.  (93), 

n  =  r0v7; 


APPLICATION  TO  BASE-LINE  WORK  343 

whence 

Ul  =  ^T=  So 
and 

Ua  = 


n      Vnl ' 
so  that  we  may  write, 

Ut-tf-^, (98) 

and 

(99) 


Example  1.  Three  measurements  of  a  base  line  under  the  same  con- 
ditions give  z  =  6717.626  ±  0.0088  ft.  and  n  =  ±  0.0152  ft.  What  is  the 
uncertainty  of  a  single  measurement  and  also  of  the  arithmetic  mean? 

n        0.0152  1 


I       6717.626       441949' 


Tr        ra        0.0088 

Ua  =  -r   = 


6717.626       763366 

Example  2.  A  base  line  of  10,000  ft.  length  is  to  be  measured  four  times 
under  conditions  which  make  the  probable  error  of  a  unit  length  of  line 
equal  ±  0.000316  ft.  What  should  be  the  uncertainty  of  each  measurement 
and  of  the  average  of  the  four  measurements? 

,.,          r0        0.000316  1 


Vl       VlOOOO"      316456' 
r0        0.000316  =    _J__ 
VlOOOO  "  632912  ' 


CHAPTER  XVI 
APPLICATION  TO  LEVEL  WORK 

207.  Unweighted  Measurements.  If  the  difference  of  ele- 
vation of  two  stations  is  measured  a  number  of  times  in  the  same 
manner,  over  the  same  length  of  line,  and  under  such  conditions 
that  the  different  determinations  may  be  regarded  as  of  equal 
weight,  thqn  (Art.  155)  the  arithmetic  mean  of  the  several  results 
is  the  most  probable  value  of  this  difference  of  elevation.  The 
probable  error  of  a  single  measurement  (Art.  173)  is  given  by  the 
formula 


0.6745 


(100) 


and  the  probable  error  of  the  arithmetic  mean  (Art.  173)  of  n 
measurements  by  the  formula 


-  1) 


(101) 


Example.     Difference  of  elevation  by  direct  observations  of  equal  weight : 


Observed  Values 

11.501ft. 

11.509ft. 

11.480ft. 

11.478ft. 

4)45. 968  ft. 

z  =  11.492ft. 


+  0.009 
+  0.017 
-0.012 
-  0.014 


0.000081 
0.000289 
0.000144 
0.000196 


0.000710 


n 


0.6745 


=  ±  0.0104  ft. 


a 

V4 

Most  probable  value  =  11.492  ±  0.0052  ft. 


344 


APPLICATION  TO  LEVEL  WORK  345 

208.  Weighted  Measurements.  If  the  difference  of  eleva- 
tion of  two  stations  is  measured  a  number  of  times  in  the  same 
manner,  and  over  the  same  length  of  line,  but  under  such  condi- 
tions that  the  different  determinations  must  be  regarded  as  of 
unequal  weight,  then  (Art.  157)  the  weighted  arithmetic  mean  of 
the  several  results  is  the  most  probable  value  of  this  difference  of 
elevation.  The  probable  error  of  a  single  measurement  of  unit 
weight  (Art.  174)  is  given  by  the  formula 


rx  =  0.6745^^5^- ,       ...     .     .     ,  v.    (102) 

the  probable  error  of  any  measurement  of  the  weight  p  (Art.  174) 
by  the  formula 

rp  =  -IL  =  0.6745  J-^-  (103) 

\  p  \p(n  —  1)  ' 

and  the  probable  error  of  the  weighted  arithmetic  mean  (Art.  174) 
by  the  formula 


Example.     Difference   of   elevation   by   direct   observations   of  unequal 


weight  : 

Observed  Values 

p 

pM 

V 

»2 

pj>2 

17.643ft. 

1 

17.643 

-0.028 

0.000784 

0.000784 

17.647ft. 

1 

17.647 

-0.024 

0.000576 

0.000576 

17.679ft. 

2 

35.358 

+0.008 

0.000064 

0.000128 

17.683ft. 

3 

53.049 

+0.012 

0.000144 

0.000432 

7)      123.697  Spy2  =  0.001920 

z  =    17.671  n  =  4 


r.=  0.6745  J0'001920    -  ±  0.0171  ft. 


=  =  ±0.0099  ft. 

V3 


0.0064  ft. 


V7 
Most  probable  value  =  17.671  ±  0.0064  ft. 


346  GEODETIC  SURVEYING 

209.  Duplicate  Lines.  In  precise  level  work  a  duplicate  line 
of  levels  is  understood  to  mean  a  line  which  is  run  twice  over  the 
same  route  with  equal  care,  but  in  opposite  directions.  The 
object  of  running  in  opposite  directions  is  to  eliminate  from  the 
mean  result  those  systematic  errors  which  are  liable  to  occur  in 
leveling,  due  to  a  rising  or  settling  of  the  instrument  or  turning 
points  during  the  progress  of  the  work.  As  explained  in  Art.  88 
the  details  of  the  work  are  so  arranged  that  these  errors  tend  to 
neutralize  each  other  to  a  large  extent  as  the  work  progresses,  so 
that  no  material  error  is  committed  by  assuming  that  the  results 
obtained  are  affected  only  by  accidental  errors.  The  most  prob- 
able value  for  the  difference  of  elevation  of  any  two  stations, 
based  on  a  duplicate  line,  is  equal  to  the  average  of  the  two  results 
furnished  by  such  a  line.  Letting  d  represent  the  discrepancy 
between  the  result  obtained  from  the  forward  line  and  that 
obtained  from  the  reverse  line,  we  thus  have 

,    d  d 

vi  =  +-2       and       v2  =   -  ^. 

Substituting  these  values  in  Eq.  (100)  and  replacing  ri  with  r, 
for  the  case  of  duplicate  lines,  we  have  for  the  probable  error 
of  a  single  determination  (forward  or  reverse)  by  a  line  of  the 
length  I, 

rt  =  0.4769v^  -  ±  0.4769d.  .     .     .   : .     (105) 

Substituting  the  same  values  in  Eq.  (101),  we  have  for  the 
probable  error  of  the  arithmetic  mean  of  the  results  obtained  by 
the  forward  and  reverse  lines, 

ra  =  db  0.3373d; (106) 

whence 

ra  (approximately)  =  db  %d (107) 

Example.     Duplicate  line  of  levels : 

Observed  Values 

29.648  ft.  0.4769  X  0.028  =  0.0134. 

29.676  ft.  0.3373  X  0.028  =  0.0094. 

d  =  0.028  ft. 

TI  =  ±  0.0134  ft.  ra  =  d=  0.0094  ft. 

Most  probable  value  =  29.662  db  0.0094  ft. 


APPLICATION  TO  LEVEL  WORK  347 

210.  Sectional  Lines.  EvefV  line  of  levels  which  includes 
one  or  more  intermediate  bench  marks  may  be  regarded  as  made 
up  of  a  series  of  sections  connecting  these  bench  marks.  In 
general  the  work  will  be  done  by  the  method  of  duplicate  leveling 
(Art.  209) ,  so  that  a  value  for  the  difference  of  elevation  of  any  two 
successive  bench  marks  (limiting  a  section)  will  be  obtained  from 
the  forward  line,  and  another  value  from  the  reverse  line.  From 
these  two  values  (Art.  209)  we  will  have  a  most  probable  value 
and  a  probable  error  for  any  given  section,  which  will  be  independ- 
ent of  all  other  sections.  In  whatever  manner  the  leveling  may 
be  done,  however,  the  subsequent  treatment  of  the  results  will  be 
the  same,  provided  the  determinations  for  each  section  are  kept 
independent.  If  ei,  €2,  .  .  .  en,  be  the  most  probable  values  for 
the  difference  of  elevation  between  the  successive  bench  marks, 
then  (Art.  168)  the  most  probable  difference  of  elevation  E 
between  the  terminal  bench  marks,  is 

E  =  ei  +  e2  .  .  .  +en  =  2e.       ...     (108) 

And  if  n,  T2,  .  .  .  rn,  be  the  probable  errors  of  the  several  values 
ei,  €2,  etc.,  then  (Art.  182)  the  probable  error  rE  for  the  total  dif- 
ference of  elevation  E,  is 


.  .     .     .     (109) 


Example.     Level  work  on  sectional  lines.     Given 

ci  =    9.116  ±  0.008  ft. 

e2  =  31.659  =t  0.031  ft. 

e3  =  22.427  d=  0.018  ft. 

E  =  9.116  +  31.659  +  22.427  =  63.202  ft. 


rE  =  V(0.008)2  +  (0.031)2  +  (0.018)2  =  ±  0.037  ft. 
Most  probable  value  E  =  63.202  ±  0.037  ft. 

211.  General  Law  of  the  Probable  Errors.  In  measuring 
the  difference  of  elevation  between  any  two  bench  marks  by  pass- 
ing (in  the  usual  way)  through  a  series  of  turning  points,  the  case 
is  essentially  one  of  sectional  measurement  (Art.  210),  in  which  the 
difference  of  elevation  for  each  section  is  measured  a  single  time, 
and  in  which  under  similar  conditions  the  average  distance 
between  turning  points  may  be  assumed  to  be  the  same  for  any 
length  of  line.  Running  a  line  of  levels  is  thus  entirely  analogous 


348  GEODETIC  SURVEYING 

to  measuring  a  base  line,  and  hence  the  same  laws  must  hold  good. 
In  accordance  with  Art.  202,  and  without  further  demonstration, 
we  may  therefore  write  as  a 

GENERAL  LAW:  Under  the  same  conditions  of  measurement 
the  probable  error  of  a  line  of  levels  varies  as  the  square  root  of  its 
length. 

From  the  considerations  on  which  this  law  is  based  it  is  evident 
that  it  is  theoretically  true  whether  the  difference  of  elevation 
assigned  to  the  terminals  of  a  line  is  the  result  of  a  single  measure- 
ment, a  number  of  measurements,  or  a  duplicate  measurement,  so 
long  as  the  lines  being  compared  are  all  identical  in  these  details. 

Example.  A  line  of  levels  10  miles  long  has  a  probable  error  of  ±  0.156  ft. 
What  is  the  theoretical  value  of  the  probable  error  for  a  line  60  miles  long, 
run  under  the  same  conditions? 


0.156  Vf§  =  0.156  >/6~=  db  0.382  ft. 
Theoretical  probable  error  of  new  line  =  ±  0.382  ft. 

212.  The    Law   of   Relative   Weight.     As  explained  in  the 
previous  article,  the  laws  derived  for  base-line  work  are  equally 
applicable  to  level  work.     In  accordance  with  Art.  203,  and  with- 
out further  demonstration,  we  may  therefore  write  as  a 

GENERAL  LAW  :  Under  the  same  conditions  of  measurement  the 
weight  of  the  result  due  to  any  line  of  levels  varies  inversely  as  the 
length  of  the  line. 

From  the  considerations  on  which  this  law  is  based  it  is  evident 
that  it  is  theoretically  true  whether  the  difference  of  elevation 
assigned  to  the  terminals  of  the  line  is  the  result  of  a  single  meas- 
urement, a  number  of  measurements,  or  a  duplicate  measurement, 
so  long  as  the  lines  being  compared  are  all  identical  in  these 
details. 

If  two  or  more  level  lines  are  run  under  different  conditions, 
they  may  be  first  weighted  so  as  to  offset  this  circumstance,  and 
then  weighted  inversely  as  their  lengths.  The  relative  weight  of 
each  line  will  then  be  the  product  of  the  weights  applied  to  it. 

213.  Probable  Error  of  a  Line  of  Unit  Length.     The  probable 
error  corresponding  to  a  given  line  of  levels  conveys  no  idea  of  the 
precision  of  the  work  unless  accompanied  by  the  length  of  the  line. 
It  is  therefore  convenient  to  reduce  the  probable  error  of  a  line  of 
levels  to  its  corresponding  value  for  a  similar  line  of  unit  length. 


APPLICATION  TO  LEVEL  WORK  349 

A  unit  of  comparison  is  thus  established  for  different  grades  or 
pieces  of  work  which  is  independent  of  the  length  of  the  lines. 
Such  a  unit  has  no  actual  existence,  but  is  purely  a  mathematical 
basis  of  comparison. 

As  explained  in  Art.  211,  the  laws  derived  for  base-line  work 
are  equally  applicable  to  level  work.  In  accordance  with  Art.  204, 
and  without  further  demonstration,  we  may  therefore  write 

rL  =  r0VL,     .......     (110) 

in  which  TL  is  the  probable  error  for  a  given  line  of  levels  of  the 
length  L,  r0  is  the  probable  error  for  a  unit  length  of  such  a  line, 
and  in  which  all  the  values  refer  to  single  measurements.  This 
equation  indicates  that  the  probable  error  of  any  given  line  of 
levels  is  equal  to  the  square  root  of  its  length  multiplied  by  the 
probable  error  for  a  unit  length  of  such  a  line.  If  r0  is  well  deter- 
mined for  given  instruments,  conditions,  and  methods,  Eq.  (110) 
informs  us  in  advance  what  is  a  suitable  probable  error  for  a 
single  line  of  levels,  and  hence  (Art.  207)  for  the  average  result 
obtained  by  re-running  such  a  line  any  number  of  times.  In 
accordance  with  this  article  the  probable  error  in  the  mean  result 
of  a  duplicate  line  is  equal  to  the  second  member  of  Eq.  (110) 
divided  by  V2.  In  any  case,  therefore,  the  level  party  knows 
whether  its  work  is  up  to  standard,  or  whether  additional  measure- 
ments are  required. 

214.  Determination  of  the  Numerical  Value  of  the  Probable 
Error  of  a  Line  of  Unit  Length.  As  explained  in  Art.  211,  the 
laws  and  rules  for  base-line  work  are  equally  applicable  to  level 
work.  The  method  of  Art.  205  is  consequently  adapted  to  the 
present  case  by  running  one  or  more  duplicate  level  lines  of 
moderate  length,  and  noting  the  length  of  line  (one  way)  and  the 
discrepancy  for  each  duplicate  line.  In  accordance  with  Eq.(97), 
and  without  further  demonstration,  we  may  therefore  write 

r0  =  0.4769A  /^?,  (111) 

\     n 

in  which  r0  is  the  probable  error  in  running  a  single  line  of  levels 
of  unit  length,  d  is  the  discrepancy  in  any  duplicate  line,  p  is 
the  weight  (reciprocal  of  the  one  way  length)  of  that  line,  and  n 
is  the  number  of  duplicate  lines. 


350 


GEODETIC  SURVEYING 


Example.     Determination  and  application  of  the  probable  error  of  a 
level  line  of  unit  length: 


Difference  of  Elevation     d 
16.298ft.    1 
16.314"     j 
16.308ft. 
16.296" 
18.540ft. 


0.016 
0.012 
0.009 
0.010 
0.015 
0.012 
0.014 
from  which  we  have 


18.549" 

18.552ft 

18.542" 

21.663ft. 

21.648" 

21.661ft. 

21  649  " 

21.664ft. 

21.650" 


0.000256 
0.000144 
0.000081 
0.000100 
0.000225 
0.000144 
0.000196 

I 
810 

810 
560 
560 

782 
782 
782 

p 

sio 

5^0 
66~0 

782 

dnr 

i 

TF2 

0.0000003160 
0.0000001778 
0.0000001446 
0.0000001786 
0.0000003085 
0.0000001841 
0.0000002506 

whence 


r0  = 


=  0.0000015602      and      n  =  7; 

±  0.000225  ft., 


0.0000015602 


which  is  therefore  the  probable  error  in  running  a  single  line  of  levels  for 

a  distance  of  one  foot  under  the  given  conditions.  For  a  single  line  of  levels 

of  any  length  L,  run  under  the  same  conditions,  the  probable  error  would 
be,  in  accordance  with  Eq.  (110), 

TL  =roVL  =  dt  0.000225 VL  ft. 
Thus  if  L  is  10,000  feet,  we  would  have 

rL  =  d=  0.000225  VToOOO  =  ±  0.0225  ft. 

And  if  such  a  line  of  levels  were  run  four  successive  times  we  should  have, 
theoretically,  for  the  probable  error  of  the  average  difference  of  elevation, 

ra  =  db  0.0225  -5-  VT=  ±  0.0113  ft. 

It  thus  becomes  known  in  advance  what   probable  error  is  to  be  expected 
under  the  given  conditions. 

215.  Multiple  Lines.  By  a  multiple  line  of  levels  is  meant  a 
set  of  two  or  more  lines  connecting  the  same  two  bench  marks 
by  routes  of  different  length.  In  order  to  find  the  most  probable 
value  for  the  difference  of  elevation  between  the  terminals  of  a 
multiple  line,  it  is  necessary  (Art.  212)  to  weight  each  constituent 
line  inversely  at  its  length.  If  the  character  of  the  work  requires 
any  of  the  lines  to  be  also  weighted  for  other  causes,  then  the 


APPLICATION  TO  LEVEL  WORK 


351 


final  weight  of  such  line  must  be_taken  as  the  product  of  its  indi- 
vidual weights.  Having  weighted  the  several  lines  as  thus  explained 
the  case  becomes  identical  with  any  case  of  weighted  measure- 
ments (Art.  208),  and  hence  the  probable  error  of  a  single  measure- 
ment of  unit  weight  is  given  by  the  formula 


=  0.6745 


(112) 


the  probable  error  of  any  of  the  lines  of  the  weight  p  by  the 
formula 

||          r,--^- 0.8745^1^,    .     .     .     (113) 

and  the  probable  error  of  the  weighted  arithmetic  mean  by  the 
formula 


-y-4=  =  0.6745 

5Miles__ 


Zpv2 
\Zp(n  - 


(114) 


.  2%  Miles 


--3>^  Miles  — 
FIG.  86. 

Example.     Three  lines  of  levels,  as  shown  in  Fig.  86,  give  the  following 
results : 

A  to  5,  5  mile  line,  +  95.659  ft. 
A  to  B,  1\  mile  line,  +  95.814  ft. 
A  to  B,  3i  mile  line,  +  95.867  ft. 

The  elevation  of  A  is  416.723  feet.     What  is  the  most  probable  value  for 
the  elevation  of  B,  and  the  probable  error  of  this  result? 

02  pv2 

0.019044   0.0038088 
0.000289   0.0001156 
0.004900   0.0014700 
Spy2  =  0.0053944 


M 

p 

pM 

V 

95.659 

0.2 

19.1318 

-  0.138 

95.814 

0.4 

38.3256 

+  0.017 

95.867 

0.3 

28.7601 

+  0.070 

Zp 

=  0.9)86.2175 

95.797 


0.9  X  2 

416.723  +  95.797  =  512.520  ft. 
Most  probable  value  for  elevation  of  B  =  512.520  ±  0.0369  ft. 


352 


GEODETIC  SURVEYING 


216.  Level  Nets.  When  three  or  more  bench  marks  are 
interconnected  by  level  lines  so  as  to  form  a  combination  of 
closed  rings,  the  resulting  figure  is  called  a  level  net.  Fig.  87 
represents  such  a  level  net,  involving  nine  bench  marks.  The 
elevation  of  any  bench  mark  is  necessarily  independent  of  any 
other  bench  mark,  but  the  differences  between  the  elevations  of 
adjacent  bench  marks  are  not  independent  quantities,  since  in 
any  closed  circuit  their  algebraic  sum  must  equal  zero.  In  the 
given  figure  there  are  evidently  fifteen  observation  equations, 
namely,  the  observed  difference  of  elevation  between  A  and  B, 
B  and  C,  etc.  But  there  are  also  seven  closed  rings,  A  BCD,  ADA, 

etc.,  forming  seven  independent  condi- 
tional equations.  Fifteen  minus  seven 
leaves  eight,  so  that  (Art.  166)  there 
can  be  but  eight  independent  quanti- 
ties involved  in  the  fifteen  observation 
equations.  The  number  of  indepen- 
dent quantities  must  evidently  be  one 
less  than  the  number  of  bench  marks, 
since  one  of  these  must  be  assumed  as 
known  or  fixed,  and  nine  minus  one 
gives  eight  as  before.  It  sometimes 
happens  that  more  than  one  line  con- 
nects the  same  two  points,  as  between 
A  and  D  in  the  fi  ure;  but  this  fact 
makes  no  difference  in  the  method  of 
computation.  Sometimes  a  point  B 

occurs  on  a  line  without  being  connected  with  any  other  point. 
Such  a  point  has  no  influence  on  the  adjustments  of  any  other 
point,  and  may  be  included  or  omitted,  as  preferred,  in  making 
such  other  adjustments.  If  omitted  ^in  adjusting  the  other 
points  its  own  most  probable  value  can  be  found  afterwards 
by  Art.  217. 

There  are  two  general  methods  of  making  the  computations 
for  the  adjustments  of  a  level  net,  each  of  which  may  be  modified 
in  a  number  of  ways.  In  the  first  method  the  most  probable 
values  are  found  for  the  several  differences  of  elevation  between 
the  bench  marks,  the  most  probable  values  for  the  elevations  of 
the  different  bench  marks  being  then  found  by  combining  these 
differences.  In  the  second  method  the  computations  are  arranged  so 


APPLICATION  TO  LEVEL  WORK 


353 


as  to  lead  directly  to  the  most  probable  values  for  the  elevations 
of  the  bench  marks,  i^n  any  case  each  of  the  connecting  lines 
must  be  properly  weighted.  If  the  lines  are  all  run  singly  they 
are  weighted  inversely  as  their  lengths  unless  some  special  con- 
dition requires  some  of  these  weights  to  be  modified.  If  all  the 
lines  are  duplicate  lines,  the  average  difference  of  elevation  in 
each  case  may  be  treated  as  if  due  to  a  single  line,  and  weighted 
inversely  as  its  length.  If  special  conditions  exist  the  weights 
must  be  made  to  correspond.  The  manner 
in  which  each  method  is  worked  out  is 
illustrated  by  the  following  example. 

Example.  Referring  to  the  level  net  indicated 
in  Fig.  88,  the  field  notes  show  the  following 
results: 

A  to  B  =  +  11.841  ft. 


B  to  C  =  - 

C  to  D  =  + 

DtoE=  - 

E  to  A  =  - 

B  to  E  =  - 

C  to  E  =  + 


5.496  ft. 
8.207  ft. 
5.720  ft. 
8.515  ft. 
3.218  ft. 
2.619  ft. 


The  figures  on  the  diagram  are  the  lengths  in  miles 
of  the  various  lines.  The  arrow-heads  show  the 
direction  in  which  each  line  was  run.  The  eleva- 
tion of  the  point  A  is  610.693  ft.  What  are  the 
most  probable  values  for  the  elevations  of  the  re- 
maining stations? 

First  method.  As  there  are  but  four  unknown 
bench  marks  (B,  C,  D,  E),  there  can  be  but  four  in- 
dependent unknowns  in  the  observation  equations. 


FIG.  88. 


As  the  lines  AB,  BC,  CD,  DE,  may  evidently  be  selected  as  the  independent 
unknowns,  we  may  write  for  the  most  probable  values  of  the  corresponding 
differences  of  elevation 

A  to  B  =  +  11.841  +  vi; 

B  to  C  =  -    5.496  +  v2; 

C  to  D  =  +    8.207  +  v3; 

D  to  E  =  -    5.720  +  V*. 

The  conditional  equations  involved  in  the  several  closed  circuits  may  then 
be  avoided  (Art.  165)  by  writing  all  the  observation  equations  in  terms  of 
these  quantities.  Writing  the  reduced  observation  equations  (Art.  163) 
directly  from  the  figure,  we  have,  by  comparison  with  'the  observed  values, 

(A  to  B)  Vl                            =       0.000  (weight  0.4); 

(B  to  C]  v2                    =       0.000  (weight  0.3); 

(C  to  D)  v3           =       0.000  (weight  0.4); 

(DtoE)  v4  =       0.000  (weight  0.3); 

(S  to  A)  -  vi  -  v2  -  v3  -  v4  =  +  0.317  (weight  0.2); 

(B  to  E)  v2  +  v3  +  v4  =  -  0.209  (weight  0.5); 

(C  to  E)  t>3  +  f4  =  +  0.132  (weight  0.5). 


354  GEODETIC  SURVEYING 

As  an  illustration  of  how  these  equations  are  formed  let  us  consider  the 
observed  line  CE. 

Most  probable  value,  C  to  D  =  +  8.207  +  03. 
Most  probable  value,  D  to  E  =  —  5.720  -f  04. 
Hence,  by  addition, 

Most  probable  value,  C  to  E  =  +  2.487  +  03  +  04. 
Observed  value,  C  to  E  =  +  2.619. 

Hence  this  observation  equation  requires 

vs  +  y4  =  +  0.132. 

No  values  of  0i,  02,  »s,  »4,  can  meet  the  requirements  of  all  the  observation 
equations,  and  hence  to  find  the  most  probable  values  of  0i,  02,  03,  04,  we 
form  the  normal  equations  in  the  usual  way,  giving, 


0.60i  +  0.2v2  +  0.203  +  0.204  =  -  0.0634; 

0.2vi  +  1.002  +  0.7%  +  0.704  =  -  0.1679; 

0.20i  +  0.702  +  1.603  +  1.204  =  -  0.1019; 

0.20i  +  0.702  +  1.2  03+  1.504  =  -  0.1019; 


whose  solution  gives 


0!  =  -  0.0556  ft.;  03  =  +  0.0092  ft.; 

02  =  -  0.1718  ft.;  04  =+  0.0123  ft.; 


whence,  for  the  most  probable  values,  we  have 


A  to  B  =  +  11.7854  ft. 

B  to  C  =  -  5.6678  ' 

C  to  D  =  +  8.2162  ' 

D  to  E  =  -  5.7077  ' 

E  to  A  =  -  8.6261  ' 

BtoE  =  -  3.1593  ' 

C  to  E  =  +  2.5085  ' 


A  =  610.693  ft. 
B  =  622.478  ' ' 
C  =  616.811" 
D  =  625.027" 
E  =  619.319" 


Second  method.  In  this  method  we  first  find  approximate  values  for  the 
unknown  elevations  by  combining  the  observed  values  in  any  convenient 
way,  thus: 

A  =  610.693  C  =  617.038  (approx.) 

+    11.841  +      8.207 

B  =  622.534  (approx.)  D  =  625.245  (approx.) 

-      5.496  -      5.720 


C  =  617.038  (approx.)  E  =  619.525  (approx.) 

and  then  write,  for  the  most  probable  values, 

A  =  610.693; 
B  =  622.534  +  v^ 
C  =  617.038  +  02; 
D  =  625.245  +  03; 
E  =  619.525  +  04. 


APPLICATION  TO  LEVEL  WORK 


355 


Substituting  these  values  in  the  observation  equations,  we  have 

A  to  B  =?  +  11.841"+  01           =  +  11.841; 

B  to  C  =  -  5.496  -  01  +  02  =  -  5.496; 

C  to  D  =  +  8.207  -  v2  +  v3  =  +  8.207; 

D  to  E  =  -  5.720  -  y3  +  y4  =  -  5.720; 

#  to  A  =  -  8.832  -04           =  -  8.515; 
BtoE=  -  3.009  -  01  +  04  =  -  3.218; 
C  toE  =  +  2.487  -  v2  +  v4  =  +  2.619. 

Reducing  and  weighting  inversely  as  the  distances,  we  have 

vi  =       0.000  (weight  0.4) ; 

-t>i+t>2  =       0.000  (weight  0.3) ; 

-  V2  +  v3  =       0.000  (weight  0.4) ; 

-  03  +  04  =       0.000  (weight  0.3) ; 

_  y4  =  -f  0.317  (weight  0.2); 

_  Vl  +  V4  =  -  0.209  (weight  0.5); 

-v2  +  04  =  +  0.132  (weight  0.5). 

Forming  the  normal  equations,  we  have 

1.20!  -  0.302  -  0.504  =  +  0.1045; 

-  0.30i  +  1.202  -  0.403  -  0.504  =  -  0.0660; 

-  0.402  +  0.703  -  0.304  =       0.0000; 
—  0.50i  -  0.502  -  0.303  +  1.504  =  -  0.1019; 

whose  solution  gives 

01  =  -  0.0556  ft.;  03  =  -  0.2182  ft.; 

02  =  -  0.2274  "  04  =  -  0.2059  " 

whence,  for  the  most  probable  values,  we  have  (as  before) 

A  =  610.693  ft.  A 

B  =  622.478  ' ' 
C  =  616.811" 
D  =  625.027  ' ' 
E  =  619.319" 

217.  Intermediate  Points.  By  an  inter- 
mediate point  is  meant  one  lying  only  on 
a  single  line  of  levels,  and  hence  having 
no  influence  on  the  general  adjustment. 
Thus  in  Fig.  89  the  bench  marks  A  and  B 
are  adjusted  as  a  part  of  the  complete  level 
net  ABCDEFG.  The  point  /  is  an  inter- 
mediate point,  having  no  influence  on  the 
general  adjustment,  but  simply  lying  be- 
tween the  dj usted  bench  marks  A  and  B. 
In  adjusting  level  net  it  s  not  necessary 
to  separate  the  intermediate  points  from  the  others,  as  the 
results  will  come  out  the  same  whether  any  or  all  of  the  inter- 
mediate points  are  omitted  or  included.  The  work  of  compu- 


FIG.  89- 


356  GEODETIC  SURVEYING 

tation  may  be  reduced,  however,  where  there  are  many  inter- 
mediate points,  by  adjusting  the  main  system  first  and  the  inter- 
mediate points  afterwards.  Referring  to  Fig.  89,  page  355, 

Let  /  be  an  intermediate  point  lying  between  the  adjusted 

bench  marks  A  and  B; 
a  =  the  distance  A  to  /; 
6  =  the  distance  7  to  B] 

d  =  the  discrepancy  between  the  line  A  B  as  run  and  the 
difference  between  the  adjusted  values  of  A  and  B 
(+  if  the  line  as  run  makes  B  too  high)  ; 
e  =  observed  change  in  elevation  from  A  to  /; 
e   —  observed  change  in  elevation  from  /  to  B] 
then 

A+e  +  e'  =  B  +  d, 
or 

e'  =  B  -  A  -  e  +  d; 
and 

/  (observed)  =  A  +  e  (weight  6)  ;  ', 

I  (observed)  =  B  —  e'  =  A  -{-  e  —  d  (weight  a); 

or,  taking  the  weighted  arithmetic  mean, 

bA  +  be  +  aA  +  ae  —  ad 


T  , 

I  (most  probable)     = 


=- 


As  7  represents  any  intermediate  point,  and  a  the  corresponding 
distance  from  the  commencement  A  of  the  given 
line,  it  follows  from  this  equation  that  the  most 
probable  values  for  any  intermediate  points  are 
les*  arrived  at  by  adjusting  for  the  discrepancy  d  in 
direct  proportion  to  the  distances  from  the  initial 
point  A.  This  law  may  be  otherwise  expressed 
by  saying  that  the  discrepancy  is  to  be  distributed 
uniformly  along  the  line  on  the  basis  of  dis- 
tance. 

Example.     In  the  line  of  levels  indicated  in  Fig.  90  the 
field  notes  show  the  following  changes  in  elevation: 

A  to  B  =  +  2.626  ft. 
B  to  C  =  -  3.483" 
CtoD  =  +6.915" 


APPLICATION  TO  LEVEL  WOEK 


357 


The  adjusted  elevations  at  A  and  D  £re 

A  =  28.655  fir. 
D  =  34.317'' 

What  are  the  most  probable  elevations  of  the  intermediate  points  B  and  C? 

28.655 
+    2.626 

—       Discrepancy  =  +  0.396  ft.       Total  distance  =  9  miles. 
31.281 
-    3.483 


0.396  X  I  =  0.088  ft. 
Station     Apparent  Elevation 

A              28.655 
B              31.281 
C              27.798 
D             34.713 

0.396  X  f  =  0.220  ft. 

Correction     Adjusted  Elevation 

0.000              28.655  ft. 
-0.088              31.193" 
-0.220              27.578" 
-0.396              34.317" 

27.798 
+    6.915 

34.713 
34.317 

+    0.396 

218.  Closed  Circuits.  By  a  closed  circuit  in  level  work  is 
meant  a  line  of  levels  which  returns  to  the  initial  point,  or,  in 
other  words,  forms  a  single  closed  ring.  The  shape  of  such  a  circuit 
is  entirely  immaterial,  whether  approxi- 
mately circular,  narrow  and  elongated, 
or  irregular  in  any  degree.  A  level  net 
is  in  general  a  combination  of  closed 
circuits,  but  these  circuits  can  not  be 
adjusted  separately,  as  they  are  not 
independent.  So  also  if  any  part  of 
the  ring  is  leveled  over  more  than  once 
it  becomes  essentially  a  level  net,  and 
must  be  adjusted  accordingly.  If,  how- 
ever, the  circuit  is  independent  of  ail 
other  work,  and  has  been  run  around  but  once  under  uniform 
conditions,  it  may  be  adjusted  by  a  simpler  process.  Referring 
to  Fig.  91, 

Let  A,  B,  C,  D,  E  be  the    bench  marks  on  an  independent 
closed  circuit; 

A  =  the  initial  bench  mark; 

a  =  distance  A-B-C  to  any  point  C; 

6  =  distance  C-D-E-A  back  to  A ; 

d  =*=  discrepancy  on  arriving  at  A  (  -f  if  too  high) ; 

e  =  observed  change  in  elevation  from  A  to  C; 

e'  =  observed  change  in  elevation  from  C  to  A  ; 
then 


FIG.  91. 


358 


GEODETIC  SURVEYING 


or 


and 

C  (observed)  =  A  +  e  (weight  b) ; 

C  (observed)  =A—e'  =  A-\-e  —  d     (weight  a) ; 

or,  taking  the  weighted  arithmetic  mean, 

•„,  u  ui  \       bA  +  be  +  aA  +  ae  —  ad 

C  (most  probable)  =  -  — j— 


=  (A  +  e)  - 


d. 


(116) 


As  C  represents  any  point  in  the  circuit,  and  a  the  corresponding 
distance  from  the  initial  point  A,  it  follows  from  this  equation 
that  the  most  probable  values  for  the  elevations  of  any  points 
B,  C,  D,  E,  etc.,  are  arrived  at  by  adjusting  the  observed  eleva- 
tions for  the  discrepancy  d  directly  as  the  respective  distances 
from  the  initial  point.  This  law  may  be  otherwise  expressed  by 
saying  that  the  discrepancy  is  to  be  distributed  uniformly  around 
the  circuit  on  the  basis  of  distance. 

Example.     In  the  closed  line  of  levels  indicated  in  Fig.  91,  page  357,  the 
field  notes  show  the  following  changes  in  elevation : 

A  to  B  =  -  2.176  ft.,  distance  =  3  miles. 

B  to  C  =  +  6.481  ft.,  distance  =  1  mile. 

C  to  D  =  —  1.712  ft.,  distance  =  2  miles. 

DtoE=  —  4.820  ft.,  distance  =  2  miles. 

E  to  A  =  +  2.017  ft.,  distance  =  3  miles. 

Given  the  elevation  of  A  as  47.913  feet,  what  are  the  adjusted  elevations 
around  the  line? 

47.913 
-    2.176 


45.737 
+    6.481 

52.218 
1.712 

50.506 
-    4.820 

45.686 
+    2.017 

47.703 
47.913 


Discrepancy  =  —  0.210  ft. 
0.210  X  T3r  =  0.057  ft 
0.210  X  IT  =  0.076  ft. 


Total  distance  =  11  miles. 
0.210  X  TT=  0.105ft. 
0.210XT8r=  0.153  ft. 


Station     Apparent  Elevation       Correction       Adjusted  Elevation 


47.913 
45.737 
52.218 
50.506 
45.686 


0.000 
+  0.057 
+  0.076 
+  0.105 
+  0.153 


47.913  ft. 
45.794" 
52.294  ' ' 
50.611" 
45.839" 


-    0.210 


APPLICATION  TO  LEVEL  WORK 


359 


219.  Branch  Lines,  Circuitsr.and  Nets.  Any  level  line,  circuit, 
or  net  that  is  independent  of  another 
system  except  for  one  common  point, 
is  called  a  branch  system.  Thus  in 
Fig.  92  the  dotted  lines  represent  the 
original  system,  A  BCD  a  branch  line, 
HKLMN  a  branch  circuit,  and  PRST  V 
a  branch  net.  In  adjusting  the  main 
system  the  results  will  be  the  same 
whether  any  or  all  of  the  branch  sys- 
tems are  included  or  omitted.  If 
there  is  much  branch  work,  however, 
the  labor  of  computation  may  be  re- 
duced by  adjusting  the  main  system 
first  and  the  branch  systems  after- 
wards. When  the  main  system  is  FIG.  92. 
adjusted  the  elevations  of  A,  H,  P,  etc., 

become  fixed  quantities  which  must  not  be  disturbed  in  adjusting 
the  branch  systems. 


TABLES 


TABLES 


TABLE    I.— CURVATURE  AND  REFRACTION   (IN  ELEVATION)* 


Dis- 
tance, 
Miles. 

Difference  in  Feet  for 

Dis- 
tance. 
Miles. 

Difference  in  Feet  for 

Curvature. 

Refraction. 

Curvature 
and 
Refraction. 

Curvature. 

Refraction. 

Curvature 
and 
Refraction. 

1 

0.7 

0.1 

0.6 

34 

771.3 

108.0 

663.3 

2 

2.7 

0.4 

2.3 

35 

817.4 

114.4 

703.0 

3 

6.0 

0.8 

5.2 

36 

864.8 

121.1 

743.7 

4 

10.7 

1.5 

9.2 

37 

913.5 

127.9 

785.6 

5 

16.7 

2.3 

14.4 

38 

963.5 

134.9 

828.6 

6 

24.0 

3.4 

20.6 

39 

1014.9 

142.1 

872.8 

7 

32.7 

4.6 

28.1 

40 

1067.6 

149.5 

918.1 

8 

42.7 

6.0 

36.7 

41 

1121.7 

157.0 

964.7 

9 

54.0 

7.6 

46.4 

42 

1177.0 

164.8 

1012.2 

10 

66.7 

9.3 

57.4 

43 

1233.7 

172.7 

1061.0 

11 

80.7 

11.3 

69.4 

44 

1291.8 

180.8 

llli.O 

12 

96.1 

13.4 

82.7 

45 

1351.2 

189.2 

1162.0 

13 

112.8 

15.8 

97.0 

46 

1411.9 

197.7 

1214.2 

14 

130.8 

18.3 

112.5 

47 

1474.0 

206.3 

1267.7 

15 

150.1 

21.0 

129.1 

48 

1537.3 

215.2 

1322.1 

16 

170.8 

23.9 

146.9 

49 

1602.0 

224.3 

1377.7 

17 

192.8 

27.0 

165.8 

50 

1668.1 

233.5 

1434.6 

18 

216.2 

30.3 

185.9 

51 

1735.5 

243.0 

1492.5 

19 

240.9 

33.7 

207.2 

52 

1804.2 

252.6 

1551.6 

20 

266.9 

37.4 

229.5 

53 

1874.3 

262.4 

1611.9 

21 

294.3 

41.2 

253.1 

54 

1945.7 

272.4 

1673.3 

22 

322.9 

45.2 

277.7 

55 

2018.4 

282.6 

1735.8 

23 

353.0 

49.4 

303.6 

56 

2092.5 

292.9 

1799.6 

24 

384.3 

53.8 

330.5 

57 

2167.9 

303.5 

1864.4 

25 

417.0 

58.4 

358.6 

58 

2244.6 

314.2 

1930.4 

26 

451.1 

63.1 

388.0 

59 

2322.7 

325.2 

1997.5 

27 

486.4 

68.1 

418.3 

60 

2402.1 

336.3 

2065.8 

28 

523.1 

73.2 

449.9 

61 

2482.8 

347.6 

2135.2 

29 

561.2 

78.6 

482.6 

62 

2564.9 

359.1 

2205.8 

30 

600.5 

84.1 

516.4 

63 

2648.3 

370.8 

2277.5 

31 

641.2 

89.8 

551.4 

64 

2733.0 

382.6 

2350.4 

32 

683.3 

95.7 

587.6 

65 

2819.1 

394.7 

2424.4 

33 

726.6 

101.7 

624.9 

66 

2906.5 

406.9 

2499.6 

*  From  Appendix  No.  9,  Report  for  1882,  United  States  Coast  and  Geodetic  Survey. 

363 


364 


GEODETIC  SURVEYING 


TABLE  II— LOGARITHMS  OF  THE  PUISSANT  FACTORS* 
(In  U.  S.  Legal  Meters) 


Lat. 

A 

B 

C 

D 

E 

F 

o 

-10 

-10 

-  10 

-  10 

-20 

—  20 

20 

8.5095499 

8.5I2I555 

0.96732 

2.1996 

5-7574 

7-772 

21 

8.5095330 

8.5121049 

o  .  99036 

2.2170 

5-77II 

7.787 

22 

8.5095155 

8.5120524 

.01252 

2.2333 

5-785I 

7.800 

23 

8  .  5094973 

8.5119979 

.  03389 

2.2485 

5-7997 

7.812 

24 

8.5094786 

8.5119416 

.05455 

2.2627 

5-8146 

7.823 

25 

8  •  5094592 

8.5118834 

.07456 

2.2759 

5  •  8300 

7-832 

26 

8.5094392 

8.5118236 

.  09399 

2.2882 

5-8458 

7.841 

2? 

8.5094187 

8.5117620 

.  11289 

2.2997 

5.8620 

7.849 

28 

8.5093977 

8.5116989 

.13131 

2.3104 

5-8785 

7.855 

29 

8.5093761 

8.5116342 

•14931 

2.3203 

5.8955 

7.861 

30 

8.5093541 

8.5115682 

.  16691 

2.3294 

5-9127 

7.866 

31 

8.5093316 

8.5115007 

.18415 

2.3379 

5  •  9304 

7.870 

32 

8.5093087 

8.5114321 

.20107 

2.3456 

5.9484 

7-873 

33 

8.5092854 

8.5113622 

.21771 

2.3527 

5  •  9667 

7-875 

34 

8.5092618 

8.5112912 

.  23408 

2.3592 

5-9853 

7.877 

35 

8.5092378 

8.5112192 

.25023 

2.3651 

6  .  0043 

7.877 

36 

8.5092135 

8.5111463 

.26616 

2.3704 

6.0237 

7.877 

37 

8.5091889 

8.5110725 

.28192 

2.3750 

6  .  0433 

7-876 

38 

8.5091640 

8.5109980 

.29752 

2.3792 

6  .  0633 

7.874 

39 

8.5091390 

8.5109228 

.31298 

2.3827 

6.0836 

7.872 

40 

8.509H37 

8.5108470 

.32832 

2.3857 

6.  1043 

7.869 

41 

8.5090883 

8.5107708 

•34357 

2.3882 

6.1253 

7.864 

42 

8.5090628 

8.5106942 

•35874 

2.3901 

6.  1467 

7.860 

43 

8-5090372 

8.5106173 

1.37385 

2.3914 

6.1684 

7-854 

44 

8.5090115 

8.5105402 

1.38893 

2.3923 

6.1905 

7.848 

45 

8.5089857 

8.5104630 

i  .  40399 

2.3926 

6.2130 

7.840 

46 

8  .  5089600 

8.5103858 

.41905 

2.3924 

6.2359 

7-832 

47 

8.5089343 

8.5103087 

•43413 

2.3917 

6.2592 

7.824 

48 

8.5089086 

8.5102317 

.44925 

2.3904 

6.2830 

7.814 

49 

8.5088831 

8.5101551 

.46442 

2.3886 

6.3071 

7.804 

50 

8.5088576 

8.5100788 

.47967 

2.3862 

6.3318 

7.792 

Si 

8.5088324 

8.5100029 

.49501 

2.3833 

6.3569 

7.780 

52 

8.5088073 

8.5099276 

.51047 

2.3799 

6.3826 

7.767 

53 

8.5087824 

8-5098530 

1.52607 

2.3759 

6.4088 

7-753 

54 

8-5087577 

8.5097791 

1.54182 

2.3713 

6-4355 

7-738 

55 

8-5087334 

8.5097060 

1.55776 

2.3661 

6  .  4629 

7-723 

56 

8.5087093 

8.5096338 

•57390 

2.3603 

6.4909 

7.706 

57 

8.5086856 

8.5095626 

•59027 

2.3539 

6.5196 

7.688 

58 

8.5086622 

8  .  5094925 

.60691 

2  •  3469 

6  .  5490 

7.669 

59 

8-5086393 

8  .  5094236 

•62383 

2.3392 

6.5792 

7-649 

60 

8.5086167 

8.509356o 

.64108 

2.3309 

6.6102 

7.627 

61 

8.5085946 

8.5092897 

.65868 

2.3218 

6.6422 

7.605 

62 

8.5085730 

8.5092248 

.67667 

2.3120 

6.6750 

7-58i 

63 

8.5085519 

8.5091614 

.  69509 

2.3014 

6.7089 

7-556 

64 

8-5085313 

8  .  5090996 

•71399 

2.2901 

6  •  7440 

7  •  529 

65 

8.5085112 

8.5090395 

•  73342 

2.2778 

6.7802 

7-501 

66 

8.5084917 

8.5089811 

•  75343 

2.2647 

6.8177 

7-471 

67 

8.5084729 

8.5089245 

.77409 

2.2506 

6.8567 

7.440 

68 

8  .  5084546 

8.5088698 

•  79546 

2.2354 

6.8972 

7.406 

69 

8.5084370 

8.5088170 

.81762 

2.2192 

6.9395 

7-371 

*  Based  on  tables  in  App.  No.  9,  Report  for  1894,  U.  S.  Coast  and  Geodetic  Survey. 


TABLES 


365 


TABLE  II.- LOGARITHMS  OF. THE  PUISSANT  FACTORS— 

(Continued) 

Log  G  =  log  diff.  for  (log  JA)  -log  diff.  for  (log  s) 


log  s 

log  difference. 

log  JX 

log  s 

log  difference. 

log  JX 

3.876 

0.0000001 

2.385 

4.922 

0.0000124 

3.431 

4.026 

002 

2.535 

4.932 

130 

3.441 

4.114 

003 

2.623 

4.941 

136 

3.450 

4.177 

004 

2.686 

4.950 

142 

3.459 

4.225 

005 

2.734 

4.959 

147 

3.468 

4.265 

006 

2.774 

4.968 

153 

3.477 

4.298 

007 

2.807 

4.976 

160 

3.485 

4  .  327 

008 

2.836 

4.985 

166 

3.494 

4.353 

009 

2.862 

4.993 

172 

3.502 

4.376 

010 

2.885 

5  .  002 

179 

3.511 

4.396 

Oil 

2.905 

5.010 

186 

3.519 

4.415 

012 

2.924 

5.017 

192 

3.526 

4.433 

013 

2.942 

5.025 

199 

3.534 

4.449 

014 

2.958 

5.033 

206 

3.542 

4.464 

015 

2.973 

5.040 

213 

3.549 

4.478 

016 

2.987 

5.047 

221 

3.556 

,4.491 

017 

3.000 

5.054 

228 

3.563 

4.503 

.,•>'--  018 

3.012 

5.062 

236 

3.571 

4.526 

020 

3.035 

5.068 

243 

3.577 

4.548 

023 

3.057 

5.075 

251 

3.584 

4.570 

025 

3.079 

5.082 

259 

3.591 

4.591 

027 

3.100 

5.088 

267 

3.597 

4.612 

030 

3.121 

5.095 

275 

3.604 

4.631 

033 

3.140 

5.102 

284 

3.611 

4.649 

036 

3.158 

5.108 

292 

3.617 

4.667 

039 

3.176 

5.114 

300 

3  .  623 

4.684 

042 

3.193 

5.120 

309 

3.629 

4.701 

045 

3.210 

5.126 

318 

3.635 

4.716 

048 

3.225 

5.132 

327 

3.641 

4.732 

052 

3.241 

5.138 

336 

3.647 

4.746 

056 

3.255 

5.144 

345 

3.653 

4.761 

059 

3.270 

5.150 

354 

3.659 

4.774 

063 

3.283 

5.156 

364 

3.665 

4.788 

067 

3.297 

5.161 

373 

3.670 

4.801 

071 

3.310 

5.167 

383 

3.676 

4.813 

075 

3.322 

5.172 

392 

3.681 

4.825 

080 

3.334 

5.178 

402 

3.687 

4.834 

084 

3.343 

5.183 

412 

3.692 

4.849 

089 

3.358 

5.188 

422 

3.697 

4.860 

094 

3.369 

5.193 

433 

3.702 

4.871 

098 

3.380 

5.199 

443 

3.708 

4.882 

103 

3.391 

5.204 

453 

3.713 

4.892 

108 

3.401 

5.209 

464 

3.718 

4.903 

114 

3.412 

5.214 

474 

3.723 

4.913 

119 

3.422 

5.219 

486 

3.728 

NOTE. — The  logarithms  in  the  above  table  require  s  to  be  expressed  in  meters  and  JX  in 
seconds  of  arc.  If  s  is  expressed  in  feet  its  logarithm  must  be  reduced  by  0.516  before  using 
in  this  table. 


366 


GEODETIC  SURVEYING 


TABLE  III.— BAROMETRIC  ELEVATIONS  * 

Containing  H  =  62737  log  — . 
B 


B. 

H. 

Dif.  for 
.01. 

B. 

H. 

Dif.  for 
.01. 

B. 

H. 

Dif.  for 
.01. 

Inches. 

Feet. 

Feet. 

Inches. 

Feet. 

Feet. 

Inches. 

Feet. 

Feet. 

11.0 

27,336 

-24.6 

14.0 

20,765 

—  19.5 

17.0 

15,476 

—  16.0 

11.1 

27,090 

14.1 

20,570 

17.1 

15,316 

24.4 

19.3 

15.9 

11.2 

26,846 

14.2 

20,377 

17.2 

15,157 

24.2 

19.  1 

15.8 

11.3 

26,604 

24.0 

14.3 

20,186 

18.9 

17.3 

14,999 

15.7 

11.4 

26,364 

23.8 

14.4 

19,997 

18.8 

17.4 

14,842 

15.6 

11.5 

26,126 

23.6 

14.5 

19,809 

18.6 

17.5 

14,686 

15.5 

11.6 

25,890 

23.4 

14.6 

19,623 

18.6 

17.6 

14,531 

15.4 

11.7 

25,656 

23.2 

14.7 

19,437 

18.5 

17.7 

14,377 

15.4 

11.8 

25,424 

14.8 

19,252 

17.8 

14;223 

23.0 

18.4 

15.3 

11.9 

25,194 

22.8 

14.9 

19,068 

18.2 

17.9 

14,070 

15.2 

12.0 

24,966 

22.6 

15.0 

18,886 

18.1 

18.0 

13,918 

15.1 

12.1 

24,740 

22.4 

15.1 

18,705 

18.0 

18.1 

13,767 

15.0 

12.2 

24,516 

22.2 

15.2 

18,525 

17.9 

18.2 

13,617 

14.9 

12.3 

24,294 

22.1 

15.3 

18,346 

17.8 

18.3 

13,468 

14.9 

12.4 

24,073 

21.9 

15.4 

18,168 

17.6 

18.4 

13,319 

14.7 

12.5 

23,854 

21.7 

15.5 

17,992 

17.5 

18.5 

13,172 

14.7 

12.6 

23,637 

21.6 

15.6 

17,817 

17.4 

18.6 

13,025 

14.6 

12.7 

23,421 

15.7 

17,643 

18.7 

12,879 

21.4 

17.3 

14.6 

12.8 

23,207 

21.2 

15.8 

17,470 

17.2 

18.8 

12,733 

14.4 

12.9 

22,995 

15.9 

17,298 

18.9 

12,589 

21.0 

17.1 

14.4 

13.0 

22,785 

20.9 

16.0 

17,127 

16.9 

19.0 

12,445 

14.3 

13.1 

22,576 

20.8 

16.1 

16,958 

16.9 

19.1 

12,302 

14.2 

13.2 

22,368 

20.6 

16.2 

16,789 

16.8 

19.2 

12,160 

14.2 

13.3 

22,162 

16.3 

16,621 

19.3 

12,018 

20.4 

16.7 

14.  1 

13.4 

21,958 

16.4 

16,454 

19.4 

11,877 

20.  1 

16.6 

14.0 

13.5 

21,757 

20.0 

16.5 

16,288 

16.4 

19.5 

11,737 

13.9 

13.6 

21,557 

16.6 

16,124 

19.6 

11,598 

19.9 

16.3 

13.9 

13.7 

21,358 

19.8 

16.7 

15,961 

16.3 

19.7 

11,459 

13.8 

13.8 

21,160 

16.8 

15,798 

19.8 

11,321 

19.8 

16.2 

13.7 

13.9 

20,962 

-19.7 

16.9 

15,636 

-16.0 

19.9 

11,184 

-13.7 

14.0 

20,765 

17.0 

15,476 

20.0 

11,047 

*  From  Appendix  No.  10,  Report  for  1881,  United  States  Coast  and  Geodetic  Survey. 


TABLES 


367 


TABLE  III.— BAROMETRIC  ELEVATIONS—  (Continued) 

30 

Containing  H  =  62737  log  — . 
B 


B. 

H. 

Dif.  for 
.01. 

B. 

H. 

Dif.  for 
.01. 

B. 

H. 

Dif.  for 
.01. 

Inches. 

Feet. 

Feet. 

Inches. 

Feet. 

Feet. 

Inches. 

Feet. 

Feet. 

20.0 

11,047 

-13.6 

23.0 

7,239 

-11.8 

26.0 

3,899 

-10.5 

20.1 

10,911 

13.5 

23.1 

7,121 

11.7 

26.1 

3,794 

10.4 

20.2 

10,776 

13.4 

23.2 

7,004 

11.7 

26.2 

3,690 

10.4 

20.3 

10,642 

13.4 

23.3 

6,887 

11.7 

26.3 

3,586 

10.3 

20.4 

10,508 

13.3 

23.4 

6,770 

11.6 

26.4 

3,483 

10.3 

20.5 

10,375 

13.3 

23.5 

6,654 

11.6 

26.5 

3,380 

10.3 

20.6 

10,242 

13.2 

23.6 

6,538 

11.5 

26.6 

3,277 

10.2 

20.7 

10,110 

13.1 

23.7 

6,423 

11.5 

26.7 

3,175 

10.2 

20.8 

9,979 

13.1 

23.8 

6,308 

11.4 

26.8 

3,073 

10.1 

20.9 

9,848 

13.0 

23.9 

6,194 

11.4 

26.9 

2,972 

10.1 

21.0 

9,718 

12.9 

24.0 

6,080 

11.3 

27.0 

2,871 

10.1 

21.1 

9,589 

12.9 

24.1 

5,967 

11.3 

27.1 

2,770 

10.0 

21.2 

9,460 

12.8 

24.2 

5,854 

11.3 

27.2 

2,670 

10.0 

21.3 

9,332 

12.8 

24.3 

5,741 

11.2 

27.3 

2,570 

10.0 

21.4 

9,204 

12.7 

24.4 

5,629 

11.1 

27.4 

2,470 

9.9 

21.5 

9,077 

12.6 

24.5 

5,518 

11.1 

27.5 

2,371 

9.9 

21.6 

8,951 

12.6 

24.6 

5,407 

11.1 

27.6 

2,272 

9.9 

21.7 

8,825 

12.5 

24.7 

5,296 

11.0 

27.7 

2,173 

9.8 

21.8 

8,700 

12.5 

24.8 

5,186 

10.9 

27.8 

2,075 

9.8 

21.9 

8,575 

12.4 

24.9 

5,077 

10.9 

27.9 

1,977 

9.7 

22.0 

8,451 

12.4 

25.0 

4,968 

10.9 

28.0 

1,880 

9.7 

22.1 

8,327 

12.3 

25.1 

4,859 

10.8 

28.1 

1,783 

9.7 

22.2 

8,204 

12.2 

25.2 

4,751 

10.8 

28.2 

1,686 

9.7 

22.3 

8,082 

12.2 

25.3 

4,643 

10.8 

28.3 

1,589 

9.6 

22.4 

7,960 

12.2 

25.4 

4,535 

10.7 

28.4 

1,493 

9.6 

22.5 

7,838 

12.1 

25.5 

4,428 

10.7 

28.5 

1,397 

9.5 

22.6 

7,717 

12.0 

25.6 

4,321 

10.6 

28.6 

1,302 

9.5 

22.7 

7,597 

12.0 

25.7 

4,215 

10.6 

28.7 

1,207 

9.5 

22.8 

7,477 

11.9 

25.8 

4,109 

10.5 

28.8 

1,112 

9.4 

22.9 

7,358 

-11.9 

25.9 

4,004 

-10.5 

28.9 

1,018 

-9.4 

23.0 

7,239 

26.0 

3,899 

29.0 

924 

368 


GEODETIC  SURVEYING 


TABLE  III.— BAROMETRIC  ELEVATIONS— Continued 

30 
Containing  H  =  62737  log  — . 


B. 

H. 

Dif.  for 
.01. 

B. 

H. 

Dif.  for 

.or. 

B. 

H. 

Dif.  for 
.01 

Inches. 

Feet. 

Feet. 

Inches. 

Feet. 

Feet 

Inches. 

Feet. 

Feet. 

29.0 

924 

29.7 

274 

30.4 

-361 

29.1 

830 

-9.4 

29.8 

182 

-9.2 

30.5 

451 

-9.0 

9.4 

9.1 

8.9 

29.2 

736 

29.9 

91 

30.6 

540 

9.3 

9.1 

8.9 

29.3 

643 

30.0 

00 

30.7 

629 

9.3 

9.1 

8.8 

29.4 

550 

30.1 

-  PI 

30.8 

717 

9.2 

9.0 

8.8 

29.5 

458 

30.2 

181 

30.9 

805 

9.2 

9.0 

—8.8 

29.6 

366 

30.3 

271 

31.0 

-893 

—9.2 

—9.0 

29.7 

274 

30.4 

-361 

TABLE  IV.— CORRECTION  COEFFICIENTS  TO  BAROMETRIC 
ELEVATIONS  FOR  TEMPERATURE  (FAHRENHEIT)  AND 
HUMIDITY  * 


w 

C 

t+t' 

C 

t+t' 

C 

0° 

-0.1025 

60° 

-0.0380 

120° 

+0.0262 

5 

-0.0970 

65 

-0.0326 

125 

+0.0315 

10 

-0.0915 

70 

-0.0273 

130 

+0.0368 

15 

-0.0860 

75 

-0.0220 

135 

+0.0420 

20 

-0.0806 

80 

-0.0166 

140 

+0.0472 

25 

-0.0752 

85 

-0.0113 

145 

+0.0524 

30 

-0.0698 

90 

-0.0058 

150 

+0.0575 

35 

-0.0645 

95 

-0.0004 

155 

+0.0626 

40 

-0.0592 

100 

+0.0049 

160 

+0.0677 

45 

-0.0539 

105 

+0.0102 

165 

+0.0728 

50 

-0.0486 

110 

+0.0156 

170 

+0.0779 

55 

-0.0433 

115 

+0.0209 

175 

+0.0829 

60 

-0.0380 

120 

+0.0262 

180 

+0.0879 

*  Based  on  Tables  I  and  IV,  Appendix  No.  10,  Report  for  1881,  United  States  Coast 
and  Geodetic  Survey. 


TABLES 


369 


TABLE   V— LOGARITHMS   OF   RADIUS   OF  CURVATURE 
(In  U.  S.  Legal  Meters) 


Latitude. 

Azimuth. 

24° 

26° 

28° 

30° 

32° 

0° 

180° 

Meridian 

6.802484 

6.802602 

6  802726 

6.802857 

6.802993 

5 

175 

185° 

355° 

2503 

2620 

2744 

2874 

3009 

10 

170 

190 

350 

2558 

2674 

2796 

2924 

3057 

15 

165 

195 

345 

2649 

2761 

2880 

3005 

3135 

20 

160 

200 

340 

2771 

2880 

2995 

3116 

3241 

30 

150 

210 

330 

3098 

3197 

3301 

3410 

3523 

40 

140 

220 

320 

3501 

3585 

3676 

3771 

3869 

50 

130 

230 

310 

6.803928 

6.803999 

6.804075 

6.804155 

6.804238 

60 

120 

240 

300 

4330 

4389 

4451 

4517 

4585 

70 

110 

250 

290 

4658 

4707 

4758 

4812 

4868 

75 

105 

255 

285 

4781 

4827 

4874 

4923 

4974 

80 

100 

260 

280 

4872 

4914 

4958 

5004 

5052 

85 

95 

265 

275 

4928 

4968 

5011 

5054 

5101 

90 

Prime  Vert. 

270 

4947 

4986 

5028 

5071 

5117 

34° 

36° 

38° 

40° 

42° 

0° 

180° 

Meridian 

6.803134 

6.803279 

6.803427 

6.803578 

6.803731 

5 

175 

185° 

355° 

3150 

3294 

3441 

3591 

3744 

10 

170 

190 

350 

3195 

3337 

3483 

3631 

3780 

15 

165 

195 

345 

3270 

3409 

3551 

3695 

3840 

20 

160 

200 

340 

3371 

3505 

3642 

3781 

3922 

30 

150 

210 

330 

3641 

3762 

3885 

4011 

4138 

40 

140 

220 

320 

3972 

4077 

4184 

4294 

4405 

50 

130 

230 

310 

6.804324 

6.804412 

6.804503 

6.804595 

6.804688 

60 

120 

240 

300 

4655 

4728 

4802 

4878 

4954 

70 

110 

250 

290 

4926 

4985 

5046 

5109 

5171 

75 

105 

255 

285 

5027 

5081 

5138 

5195 

5253 

80 

100 

260 

280 

5102 

5153 

5206 

5259 

5313 

85 

95 

265 

275 

5148 

5197 

5247 

5299 

5350 

90 

Frime  Vert, 

270 

5164 

5212 

5261 

5312 

5363 

44° 

46° 

48° 

50° 

52° 

0° 

180° 

Meridian 

6.803885 

6.804040 

6.804194 

6.804347 

6.804498 

5 

175 

185° 

355° 

3897 

4050 

4204 

4356 

4506 

10 

170 

190 

350 

3931 

4082 

4233 

4383 

4531 

15 

165 

195 

345 

3987 

4135 

4282 

4428 

4573 

20 

160 

200 

340 

4064 

4206 

4348 

4489 

4629 

30 

150 

210 

330 

4267 

4396 

4524 

4652 

4778 

40 

140 

220 

320 

4516 

4628 

4740 

4851 

4960 

50 

130 

230 

310 

6.804782 

6.804876 

6.804970 

6.805063 

6.805155 

60 

120 

240 

300 

5030 

5109 

5186 

5262 

5338 

70 

110 

250 

290 

5234 

5298 

5362 

5425 

5487 

75 

105 

255 

285 

5312 

5369 

5428 

5486 

5543 

80 

100 

260 

280 

5368 

5422 

5477 

5531 

5584 

85 

95 

265 

275 

5402 

5455 

5507 

5559 

5610 

90 

Prime  Vert. 

270 

5414 

5465 

5517 

5568 

5618 

370 


GEODETIC  SURVEYING 


TABLE  VI.— LOGARITHMS  OF  RADIUS  OF  CURVATURE 

(In  feet) 


x 

Latitude. 

Azimuth. 

28° 

30° 

32° 

34° 

36° 

0°|  180° 

Meridian 

7.318711 

7.318841 

7.318978 

7.319118 

7.319263 

5 

175 

185° 

355° 

8728 

8858 

8993 

9134 

9278 

10 

170 

190 

350 

8780 

8908 

9041 

9179 

9321 

15 

165 

195 

345 

8864 

8989 

9119 

9254 

9393 

20 

160 

200 

340 

8979 

9100 

9225 

9355 

9489 

30 

150 

210 

330 

9285 

9394 

9507 

9625 

9746 

40 

140 

220 

320 

9660 

9755 

9853 

9956 

320061 

50 

130 

230 

310 

7.320059 

7.320139 

7.320222 

7.320308 

7.320396 

60 

120 

240 

300 

0435 

0501 

0569 

0639 

0712 

70 

110 

250 

290 

0742 

0796 

0852 

0910 

0969 

75 

105 

255 

285 

0858 

0907 

0958 

1011 

1065 

80 

100 

260 

280 

0942 

0988 

1036 

1086 

1137 

85 

95 

265 

275 

0995 

1038 

1085 

1132 

1181 

90 

Prime  Vert. 

270 

1012 

1055 

1101 

1148 

1196 

38° 

40° 

42° 

44° 

46° 

0° 

180° 

Meridian 

7.319412 

7.319562 

7.319715 

7.319869 

7.320024 

5 

175 

185° 

355° 

9425 

9575 

9728 

9881 

0034 

10 

170 

190 

350 

9467 

9615 

9764 

9915 

0066 

15 

165 

195 

345 

9535 

9679 

9824 

9971 

0119 

20 

160 

200 

340 

9626 

9765 

9906 

320048 

0190 

30 

150 

210 

330 

9869 

9995 

320122 

0251 

0380 

40 

140 

220 

320 

320168 

320278 

0389 

0500 

0612 

50 

130 

230 

310 

7.320487 

7.320579 

7.320672 

7.320766 

7.320860 

60 

120 

240 

300 

0786 

0862 

0938 

1014 

1093 

70 

110 

250 

290 

1030 

1093 

1155 

1218 

1282 

75 

105 

255 

285 

1122 

1179 

1237 

1296 

1353 

80 

100 

260 

280 

1190 

1243 

1297 

1352 

1406 

85 

95 

265 

275 

1231 

1283 

1334 

1386 

1439 

90 

Prime  Vert. 

270 

1246 

1296 

1347 

1398 

1449 

TABLE  VII.— CORRECTIONS  FOR  CURVATURE  AND  REFRACTION 
IN  PRECISE  SPIRIT  LEVELING 


Distance. 

Correction 
to  Rod 
Reading. 

Distance. 

Correction 
to  Rod 
Reading. 

Distance. 

Correction 
to  Rod 
Reading. 

Meters 

mm. 

Meters. 

mm. 

Meters. 

mm. 

0  to    27 

0.0 

100 

-0.68 

200 

-2.73 

28  to    47 

-0.1 

110 

-0.83 

210 

-3.01 

48  to    60 

-0.2 

120 

-0.98 

220 

-3.31 

61  to    72 

-0.3 

130 

-1.15 

230 

-3.61 

73  to    81 

-0.4 

140 

-1.34 

240 

-3.94 

82  to    90 

-0.5 

150 

-  1  .  54 

250 

-4.27 

91  to    98 

-0.6 

160 

-1.75 

260 

-4.62 

99  to  105 

-0.7 

170 

-1.97 

270 

-4.98 

106  to  112 

-0.8 

180 

-2.21 

280 

-5.36 

113  tollS 

-0.9 

190 

-2.47 

290 

-5.75 

TABLES 
TABLE  VIIL— MEAN  ANGULAR  REFRACTION 


371 


Apparent 
Altitude. 

Refraction. 

Apparent 
Altitude. 

Refraction. 

Apparent 
Altitude. 

Refraction. 

Apparent 
Zenith 
Distance. 

o         / 

/           // 

o 

/          // 

o 

i          ii 

0 

0    00 

34     54.1 

10 

5     16.2 

50 

0    48.4 

40 

10 

32     49.2 

11 

4    48.5 

51 

0    46.7 

39 

20 

30     52.3 

12 

4    25.0 

52 

0     45.1 

38 

30 

29     03.5 

13 

4    04.9 

53 

0    43.5 

37 

40 

27     22.7 

14 

3     47.4 

54 

0    41.9 

36 

50 

25     49.8 

15 

3     32.1 

55 

0    40.4 

35 

1     00 

24     24.6 

16 

3     18.6 

56 

0    38.9 

34 

10 

23     06.7 

17 

3    06.6 

57 

0    37.5 

33 

20 

21     55.6 

18 

2     55.8 

58 

0    36.1 

32 

30 

20     50.9 

19 

2    46.1 

59       - 

0    34.7 

31 

40 

19     51.9 

50 

18     58.0 

20 

2     37.3 

60 

0    33.3 

30 

21 

2     29.3 

61 

0    32.0 

29 

2    00 

18     08.6 

22 

2     21.9 

62 

0     30.7 

28 

10 

17     23.0 

23 

2     15.2 

63 

0     29.4 

27 

20 

16     40.7 

24 

2     08.9 

64 

0    28.2 

26 

30 

16     00.9 

40 

15     23.4 

25 

2     03.2 

65 

0    26.9 

25 

50 

14    47.8 

26 

57.8 

66 

0     25.7 

24 

27. 

52.8 

67 

0     24.5 

23 

3     00 

14     14.6 

28 

48.2 

68 

0    23.3 

.22 

10 

13     43.7 

29 

43.8 

69 

0    22.2 

21 

20 

13     15.0 

30 

12     48.3 

30 

39.7 

70 

0    21.0 

20 

40 

12     23.7 

31 

35.8 

71 

0     19.9 

19 

50 

12     00.7 

32 

32.1 

72 

0     18.8 

18 

33 

28.7 

73 

0     17.7 

17 

4    00 

11     38.9 

34 

25.4 

74 

0     16.6 

16 

10 

11     18.3 

20 

10     58.6 

35 

22.3 

75 

0     15.5 

15 

30 

10     39.6 

36 

19.3 

76 

0     14.5 

14 

40 

10    21.2 

37 

16.5 

77 

0     13.4 

13 

50 

10    03.3 

38 

13.8 

78 

0     12.3 

12 

39 

11.2 

79 

0     11.2 

11 

5     00 

9     46.5 

30 

9    01.9 

40 

08.7 

80 

0     10.2 

10 

41 

06.3 

81 

0,   09.1 

9 

6    00 

8     23.3 

42 

04.0 

82 

0    08.1 

8 

30 

7     49.5 

43 

01.8 

83 

0    07.1 

7 

44 

0     59.7 

84 

0    06.1 

6 

7    00 

7     19.7 

30 

6     53.3 

45 

0    57.7 

85 

0    05.1 

5 

46 

0    55.7 

86 

0    04.1 

4 

8    00 

6     29.6 

47 

0     53.8 

87 

0    03.0 

3 

30 

6     08.4 

48 

0     51.9 

88 

0    02.0 

2 

49 

0     50.2 

89 

0    01.0 

1 

9    00 

5     49.3 

30 

5     32.0 

50 

0     48.4 

90 

0    00.0 

0 

372 


GEODETIC  SURVEYING 


TABLE   IX.— ELEMENTS  OF  MAP  PROJECTIONS 


Lat. 

4> 

Logarithms  (U.  S.  Legal  Meters). 

1°  in  Meters. 

Logarithm 
(1  —  e-sin2  <£)• 

(-10) 

R 

N 

r 

Latitude. 
(0-30'  to 
0  +  30') 

Longitude. 
(On  Par.  of 
Latitude.) 

20° 

6  .  8022696 

6.8048752 

6.7778610 

110700 

104650 

9.9996560 

22 

3727 

9096 

.7720755 

726 

103265 

5873 

24 

4835 

9465 

.7656767 

754 

101755 

5134 

26 

6015 

9859 

.7586461 

785 

100121 

4347 

28 

7262 

6.8050274 

.7509623 

816 

98365 

3516 

30 

6.8028569 

6.8050710 

6.7426016 

110850 

96489 

9.9992645 

32 

9930 

1164 

.  7335369 

884 

94496 

1738 

34 

6.8031339 

1633 

.7237375 

920 

92388 

0798 

36 

2788 

2116 

.7131692 

957 

90167 

9.9989832 

38 

4271 

2611 

.7017932 

995 

87836 

8843 

40 

6.8035781 

6.8053114 

6  .  6895654 

111034 

85397 

9  .  0987837 

42 

7309 

3623 

.  6764358 

073 

82854 

6818 

44 

8849 

4136 

.  6623477 

112 

80209 

5792 

46 

6.8040393 

4651 

.6472364 

152 

77466 

4762 

48 

1934 

5165 

.6310274 

191 

74629 

3735 

50 

6  .  8043463 

6  .  8055675 

6.6136350 

111231 

71699 

9.9982715 

52 

4975 

6178 

.  5949598 

269 

68681 

1708 

54 

6460 

6674 

.5748861 

307 

65579 

0717 

56 

7913 

7158 

.5532775 

345 

62396 

9.9979749 

58 

9326 

7629 

.5299726 

381 

59136 

8807 

60 

6.8050691 

6  .  8058084 

6.5047784 

111416 

55803 

9.9977897 

Element 

Coordinates  of  Developed  Arcs. 

Lat 

of 
Tangent 

X 

V 

*  ' 

Cone. 

for  1°  of  Long. 

for  n°  cf  Long. 

for  1°  of  Long. 

for  n°. 

Miles. 

Miles. 

Meters. 

Value  f  or  (1°)  X 

Miles. 

Meters. 

(1°)  X 

20° 

10893 

65.03 

104649 

n.cos(0.197n°) 

0.1941 

312.3 

n2 

22 

9814 

64.17 

103264 

n-cos  (0.216n°) 

0.2098 

337.6 

n2 

24 

8907 

63.23 

101754 

n-cos  (0.235n°) 

0  .  2244 

361.2 

n2 

26 

8131 

62.21 

100120 

n-cos  (0.253n°) 

0  .  2380 

383.0 

n2 

28 

7459 

61.12 

98364 

n-cos  (0.271n°) 

0  .  2504 

403.0 

n2 

30 

6870 

59.95 

96488 

n-cos(0.288n°) 

0.2616 

421.0 

n2 

32 

6349 

58.72 

94495 

n-cos  (0.305n°) 

0.2715 

437.0 

n2 

34 

5882 

57.41 

92386 

n-cos  (0.322n°) 

0.2801 

450.8 

n2 

36 

5461 

56.03 

90165 

n-cos  (0.339n°) 

0.2874 

462.5 

n2 

38 

5079 

54.58 

87834 

n-cos  (0.355n°) 

0  .  2932 

471.9 

n2 

40 

4730 

53.06 

85395 

n-cos(0.371n°) 

0.2976 

479.0 

n2 

42 

4408 

51.48 

82852 

n-cos(0.386n°) 

0  .  3006 

483.8 

n2 

44 

4111 

49.84 

80207 

n-cos  (0.400n°) 

0.3021 

486.2 

n2 

46 

3834 

48.13 

77464 

n-cos  (0.414n°) 

0  .  3022 

486.3 

n2 

48 

3575 

46.37 

74627 

n-cos  (0.428n°) 

0  .  3007 

484.0 

n2 

50 

3332 

44.55 

71697 

n-cos(0.441n°) 

0.2978 

479.3 

n2 

52 

3103 

42.68 

68679 

n-cos(0.454n°) 

0.2935 

472.3 

n2 

54 

2886 

40.75 

65577 

n-cos  (0.466n°) 

0.2877 

463.0 

n2 

56 

2679 

38.77 

62394 

n-cos  (0.478n°) 

0.2805 

451.4 

n2 

58 

2483 

36.74 

59134 

n-cos  (0.489n°) 

0.2719 

437.6 

n2 

60 

2294 

34.67 

55801 

n-cos  (0.499n°) 

0.2620 

421.7 

n2 

TABLES 


TABLE  X.— CONSTANTS    AND    THEIR    LOGARITHMS 


373 


General  Constants. 

Number. 

Logarithm. 

71   

3.141592654 
0.318309886 
9.869604401 
0.101321184 
1.772453851 
0.564189584 

57.29577951 
3437.746771 
206264.8062 

0.017453293 
0.017452406 
0  .  000290888 
0  .  000290888 
0  .  000004848 
0  .  000004848 

2.718281828 
0.434294482 
0.434294482 
2.302585093 

3.2808693.. 
3.280833333 

0.621369949 
1609.347219 

0.4769363.  . 
0.6744897.. 

0.4971498727 
9.5028501273 
0.9942997454 
9  .  0057002546 
0  .  2485749363 
9.7514250637 

1.7581226324 
3  .  5362738828 
5.3144251332 

8.2418773676 
8.2418553184 
6.4637261172 
6.4637261109 
4  .  6855748668 
4.6855748668 

0.4342944819 
9.6377843113 
9.6377843113 
0.3622156887 

0.5159889297 
0,5159841687 

9  .  7933502462 
3  .  2066497538 

9.6784604... 
9.8289754... 

-10 
-10 
-10 

-10 
-10 
-10 
-10 
-10 
-10 

-10 
-10 

-10 

-10 
-10 

1 

7T 

rr2  

1 

7:2  
VTT  

1 

V'JT 
Degrees  in  a  radian  

Minutes  in  a  radian  .... 

Seconds  in  a  radian  .  .  . 

Arc  1°.  . 

Sin  1°  

Arc  1'  

Sin  1'  

Arc  1"  .... 

Sin  1"  

Base  of  natural  logarithms  (e)  .  . 

Modulus  of  common  logarithms  (M) 

Common  log  x  -^natural  log  x 

Natural  log  x  -f-  common  log  x 

1  Clarke  meter  =  3.2808693  ft.  . 

1  U.  S.  legal  meter  =  3.2808333  +  ft  

1  kilometer  =  five-eighths  mile,  nearly  .  .  . 
1  statute  mile  =  1609  +  meters  

Probable  error  function  hr  

Probable  error  constant  hr  ^2  

Geodetic  Constants. 
(Clarke's  1866  Spheroid.) 

Logarithms. 

U.  S.  Legal  Meters. 

Feet. 

Semi-major  axis  =  a 

6.804703? 
6  .  8032285 
9  .  9985252 
6  .  8039665 

7.5302093 
8.9152513 
7.8305026 
9.9970504 
6.8017537 
6.8061781 

- 

10 
10 
10 
10 
10 

7.3206875 
7.3192127 
9  .  9985252 
7.3199507 

7.5302093 
8.9152513 
7.8305026 
9.9970504 
7.3177379 
7.3221623 

-10 
-10 
10 

Semi-minor  axis  =  6  —  a  ^1  —  ez 

Ratio  of  axes=293  98  —  294  98 

Mean  radius 

Ellipticity  =  —  —  =  e 

a 

-10 
-10 

&2 

—  =  1  —  e2 

a 
a2           a 

b        VT^-e- 

BIBLIOGRAPHY 


REFERENCES   ON   GEODETIC   SURVEYING 

Adjustment  of  Observations,  Wright  and  Hayford.     D.  Van  Nostrand  & 

Co.,  New  York,  1904. 

Elements  of  Geodesy,  Gore.    John  Wiley  &  Sons,  Naw  York,  1893. 
Gillespie's  Higher  Surveying,  Staley.     D.  Appleton  &  Co.,  New  York,  1897. 
Johnson's  Theory  and  Practice  of  Surveying,  Smith.     John  Wiley  &  Sons, 

New  York,  1910. 
Manual  of  Spherical  and  Practical  Astronomy,  Chauvenet.     J.  B.  Lippin- 

cott  &  Co.,  Philadelphia,  1885. 
Practical  Astronomy   as   Applied   to   Geodesy   and   Navigation,    Doolittle. 

John  Wiley  &  Sons,  New  York,  1893. 
Precise   Surveying   and    Geodesy,    Merriman.     John   Wiley   &   Sons,    New 

York,  1899. 
Principles  and  Practice  of  Surveying,  Breed  arid  Hosmer.     John  Wiley  & 

Sons,  New  York,  1906. 
Text  Book  of  Field  Astronomy  for  Engineers,  Comstock.     John  Wiley  & 

Sons,  New  York,  1902. 
Text  Book  of  Geodetic  Astronomy,  Hayford.     John  Wiley  &  Sons,  New  York, 

1898. 
Text  Book  on  Geodesy  and  Least  Squares,  Crandall.     John  Wiley  &  Sons, 

New  York,  1907. 
Geodesic  Night  Signals,  Appendix  No.  8,  Report  for  1880,  U.  S.  Coast  and 

Geodetic  Survey. 
Field  Work  of  the  Triangulation,  Appendix  No.  9,  Report  for  1882,  U.  S. 

Coast  and  Geodetic  Survey. 
Observing  Tripods  and  Scaffolds,  Appendix  No.  10,  Report  for  1882,  U.  S. 

Coast  and  Geodetic  Survey. 
Geodetic  Reconnaissance,  Appendix  No.  10,  Report  for  1885,  U.  S.  Coast 

and  Geodetic  Survey. 
Relation  of  the  Yard  to  the  Meter,  Appendix  No.  16,  Report  for  1890,  U.  S. 

Coast  and  Geodetic  Survey. 
Fundamental  Standards  of  Length  and  Mass,  Appendix  No.  6,  Report  for 

1893,  U.  S.  Coast  and  Geodetic  Survey. 
Perfected  Form  of  Base  Apparatus,  Appendix  No.  17,  Report  for  1880,  U.  S. 

Coast  and  Geodetic  Survey. 

374 


BIBLIOGRAPHY  375 

Description  of  a  Compensating  Base  Apparatus,  Appendix  No.  7,  Report 

for  1882,  U.  S.  Coast  and  Geodetic  Survey. 
The  Eimbeck  Duplex  Base-bar,  Appendix  No.  11,  Report  for  1897,  U.  S. 

Coast  and  Geodetic  Survey. 
Measurement  of  Base  Lines  (Jaderin  Method)  with  Steel  Tapes  and  with 

Steel  and  Brass  Wires,  Appendix  No.  5,  Report  for  1893,  U.  S.  Coast 

and  Geodetic  Survey. 
Measurement  of  Base  Lines  with  Steel  and  Invar  Tapes,  Appendix  No.  4, 

Report  for  1907,  U.  S.  Coast  and  Geodetic  Survey. 
Run  of  the  Micrometer,  Appendix  No.  8,  Report  for  1884,  U.  S.  Coast  and 

Geodetic  Survey. 
Synthetic  Adjustment  of  Triangulation  Systems,  Appendix  No.  12,  Report 

for  1892,  U.  S.  Coast  and  Geodetic  Survey. 
Formulas  and  Tables  for  the  Computation  of  Geodetic  Positions,  Appendix 

No.  9,  Report  for  1894,  and  Appendix  No.  4,  Report  for  1901,  U.  S. 

Coast  and  Geodetic  Survey. 
Barometric  Hypsometry,  Appendix  No.  10,  Report  for  1881,  U.  S.  Coast 

and  Geodetic  Survey. 
Transcontinental  Line  of  Leveling  in  the  United  States,  Appendix  No.  11, 

Report  for  1882,  U.  S.  Coast  and  Geodetic  Survey. 
Self-registering  Tide  Gauges,  Appendix  No.  7,  Report  for  1897,  U.  S.  Coast 

and  Geodetic  Survey. 
Precise  Leveling  in  the  United  States,  Appendix  No.  8,  Report  for  1899,  and 

Appendix  No.  3,  Report  for  1903,  U.  S.  Coast  and  Geodetic  Survey. 
Variations  in  Latitude,  Appendix  No.  13,  Report  for  1891,  Appendix  No.  1, 

Report  for  1892,  Appendix  No.  2,  Report  for  1892,  and  Appendix  No. 

11,  Report  for  1893,  U.  S.  Coast  and  Geodetic  Survey. 
Tables  of  Azimuth  and  Apparent  Altitude  of  Polaris,  Appendix  No.   10, 

Report  for  1895,  U.  S.  Coast  and  Geodetic  Survey. 
Determination  of  Time,  Latitude,  Longitude,  and  Azimuth,  Appendix  No.  7, 

Report  for  1898,  U.  S.  Coast  and  Geodetic  Survey. 
A  Treatise  on  Projections,  U.  S.  Coast  and  Geodetic  Survey,  1882. 
Tables   for   the   Polyconic   Projection    of   Maps    (Clarke's    1866   Spheroid), 

Appendix  No.  6,  Report  for  1884,  U.  S.  Coast  and  Geodetic  Survey. 
Geographical  Tables  and  Formulas,  U.  S.  Geological  Survey,  1908. 
Bibliography  of  Geodesy  (Gore),  Appendix  No.  8,  Report  for  1902,  U.  S. 

Coast  and  Geodetic  Survey. 

REFERENCES  ON  METHOD  OF  LEAST  SQUARES. 

Manual  of  Spherical  and  Practical  Astronomy,  Chauvenet.     J.  B.  Lippincott 

Co.,  Philadelphia,  1885. 
Approximate  Determination  of  Probable  Error,  Appendix  No.  13,  Report 

for  1890,  U.  S.  Coast  and  Geodetic  Survey. 
Theory  of  Errors  and  Method  of  Least  Squares,  Johnson.     John  Wiley  & 

Sons,  New  York,  1893. 
Practical   Astronomy   as   Applied   to    Geodesy   and   Navigation,    Doolittle. 

John  Wiley  &  Sons,  New  York,  1893. 


376  BIBLIOGRAPHY 

Piecise    Surveying   and   Geodesy,    Merriman.     John   Wiley   &   Sons,    New 

York,  1899. 
Adjustment  of  Observations,  Wright  and  Hayford.     D.  Van  Nostrand  & 

Co.,  New  York,  1904. 
Text  Book  on  Geodesy  and  Least  Squares,  Crandall.     John  Wiley  &  Sons, 

New  York,  1907. 


INDEX 


Aberration  of  light' (diurnal) 213 

Absolute  length,  correction  for 36 

Absolute  locations \ 4 

Accidental  errors: 

laws  of 252 

nature  of 247 

theory  of 252-265 

Accuracy  attainable  in 

angle  measurements 78 

barometric  leveling 129 

base-line  measurement 45 

closing  triangles 102 

precise  spirit  leveling 161 

trigonometric  leveling 139 

Adjustment  of 

angle  measurements 81,  100,  312-332 

base-line  measurements • 333 

level  work 160,  344-359 

observations 3,  241-359 

quadrilaterals 90-100,  327 

triangles 89,  322-326 

Adjustments  of 

Coast  Survey  precise  level 155-156 

direction  instrument 65 

European  type  of  precise  level 146-152 

repeating  instrument 59 

Alignment  corrections: 

horizontal 40 

vertical 42 

Alignment  curve 64 

Altazimuth  instrument 48,  51 

Altitude 167 

American  Ephemeris 164 

Aneroid  barometer 126,  127 

377 


378  INDEX 

PAGE 

Angles: 

accuracy  of  measurements 78 

adjustment  of 81,  100,  312-332 

eccentric 75 

exterior  and  interior 53 

instruments  for  measuring 47,  52,  60 

measurement  of 47-80 

Apparent  time 165 

Arcs,  elliptic 108 

Arithmetic  mean • 244 

Associations,  geodetic 1 

Astronomical  determinations 163-226 

See  also  Azimuth,  Latitude,  Longitude,  and  Time. 

Azimuth 4,  109,  167,  203 

astronomical 204 

geodetic 204 

lines,  planes  and  sections Ill 

marks 204 

periodic  changes  in 226 

Azimuthal  angles 109,  117 

Azimuth  determinations 203-226 

approximate 214 

at  sea 225 

by  meridian  altitudes  of  sun  or  stars 205 

by  observations  on  circumpolar  stars 207-225 

direction  method 215 

fundamental  formulas .  . 208 

micrometric  method 221 

repeating  method 218 

Back  azimuth 109,  113,  122 

errors 122 

Barometers,  aneroid  and  mercurial 126,  127 

Barometric  leveling 125,  126-130 

Base-bars 24-29 

compensating 26 

Eimbeck  duplex 26 

general  features  of 25 

standardizing 33 

thermometric 26 

tripods  for 27 

Base-line  measurements 24-46,  333-343   t 

accuracy  of 45 

adjustment  of 333 

check  bases 5 

corrections  required 24,  35-44 

duplicate  lines 334 


INDEX  379 

i 

PAGE 

Base-line  measurements-^ (continued) 

gaps,  computing  length  of 44 

general  law  of  probable  error 336 

law  of  relative  weight 337 

probable  error  of 65 

probable  error  of  lines  of  unit  length 338,  339 

sectional  lines 335 

standardizing  bars  and  tapes 33 

with  base-bars 24-29 

with  steel  and  brass  wires 32 

with  steel  and  invar  tapes 30-32 

uncertainty  of '. 46,  342 

Bessel's  solution  of  geodetic  problem 118 

Bessel's  spheroid 106 

Bibliography 374 

Board  signals 20 

Bonne's  map  projection 238 

Celestial  sphere 166 

Chance,  laws  of 248 

Changes,  periodic: 

in  azimuth 226 

in  latitude 196 

in  longitude 203 

Check  bases 5 

Chronograph 184 

Circumpolar  stars 190,  207 

Clarke's  spheroid 106 

Clarke's  solution  of  geodetic  problem: 

direct 116 

inverse 118 

Closed  level  circuits .   160,  357 

Closing  the  horizon 53,  313,  315 

Coast  and  Geodetic  Survey,  United  States / 1 

papers  of 1 

precise  level 153 

Coefficient  of  refraction 138 

Co-functions : 

altitude 

declination 167 

latitude I67 

Comparator 34 

Compensating  base-bars 26 

Computation  of  geodetic  positions 103-124 

Bessel's  solution 118 

Clarke's  solution 

Helmert's  solution .  .                   118 


380  INDEX 

PAGE 

Computations  of  geodetic  positions — (continued} 

inverse  problem , 118 

Puissant's  solution 113 

Computed  quantities: 

most  probable  values  of 296 

probable  errors  of 306-311 

Conditional  equations 284 

Conditioned  quantities: 

definition  of 242,  284 

most  probable  values  of 284-295 

probable  errors  of 304 

Convergence  of  meridians 88,  111 

Corrections  in  base-line  work 24,  35-44 

Correlative  equations 290 

Cross-section  of  tapes 38 

Culmination,  meaning  of 190 

Curvature  and  refraction  (in  elevation) 12 

Declination 167 

Degree,  length  of: 

meridian 228 

parallel  of  latitude 228 

Dependent  equations 284 

Dependent  quantities: 

definition  of 242,  284 

most  probable  values  of 284-295 

probable  errors  of 304 

Deviation  of  plumb  line 124 

Dip  of  horizon 184 

Direction  instrument 47,  50,  60 

adjustments  of 71 

Direct  observations 243 

Distances,  polar  and  zenith 167 

Diurnal  aberration .  .  , 213 

Duplex  base-bars 26 

Duplicate  base  lines 334 

Duplicate  level  lines 160,  346 

Earth,  figure  of: 

general  figure > 104 

practical  figure 106 

precise  figure 105 

Eccentric  signals 20,  78 

stations 75 

Eimbeck  duplex  base-bar 26 

Elevation  of  stations 62 

Ellipsoid,  definition  of 105 


INDEX  381 


•  • 


Elliptic  arcs f. • 1Q8 

Elongation,  definition  of 208 

Ephemeris,  American 164 

Equation  of  time 165 

Equations : 

conditional 284 

correlative ,-.••.• 290 

dependent 284 

normal 273,  275 

observation , 271 

probabib'ty , , 257 

reduced  observation , 281 

Errors : 

classification  of ' 245,  247 

facility  of • 255 

in  precise  leveling , 143 

laws  of 252 

probability  of 256 

theory  of : .- 252-265 

types  of 254 

European  precise  level 141,  145 

adjustments  of 146-152 

Exterior  angles 53 

Figure  adjustment •  81,  87,  100,  312,  321-332 

Figure  of  earth : 

analytical  considerations 110 

constants  of > 106 

general  figure v 104 

geometrical  considerations 106 

practical  figure . 106 

precise  figure 105 

Filar  micrometer: 

description  of 66 

reading  the  micrometer 67 

run  of  the  micrometer 68 

Flattening  of  the  earth's  poles 104,  105 

Foot  pins  and  plates 158 

Gaps  in  base  lines 44 

Geodesic  line 109 

Geodesy: 

definition  of 1 

history  of 1 

scope  of 2 

Geodetic  associations 1 

Geodetic  leveling 125-162 


382  INDEX 

PAGE 

Geodetic  map  drawing 227-240 

Geodetic  positions,  computation  of 103-124 

Geodetic  quadrilateral 7,  90,  327 

Geodetic  surveying 1-240 

Geodetic  work  in  the  United  States 1 

Geoid,  definition  of 106 

Geometric  mean 244 

Harrebow-Talcott  latitude  method 193 

Heat  radiation 47 

Height  of  stations 17 

Heliotropes 21 

Helmert's  solution  of  geodetic  problem 118 

History  of  plane  and  geodetic  surveying 1 

Horizontal  alignment 40 

Hour  angle 164,  167 

Independent  quantities : 

definition  of 241 

most  probable  values  of 266-283 

probable  errors  of 300-304 

Indirect  observations 243 

Instruments,  geodetic;  see  Angles,  Astronomical  determinations,  Base- 
line measurements  and  Geodetic  leveling. 

Interior  angles 53 

Intermediate  points  in  leveling 160,  217 

International  Geodetic  Association 1 

Intel-visibility  of  stations 11,  14 

Invar  tapes 32 

Inverse  geodetic  problem 118 

Jaderin  base-line  methods: 

with  tapes 31 

with  wires 32 

Latitude 109,  167,  186 

astronomical 186 

geocentric 187 

geodetic 186 

locating  a  parallel  of 120 

periodic  changes  in 196 

Latitude  determinations 188-196 

at  sea 196 

by  circumpolar  culminations 190 

by  Harrebow-Talcott  method 193 

by  meridian  altitudes  of  sun 188 

by  prime- vertical  transits 192 

by  zenith  telescope 193 


INDEX  383 

PAGE 

Law  of 

coefficients  in  correlative  equations 294 

coefficients  in  normal  equations 280 

facility  of  error 257 

Laws  of 

chance 248-250 

errors 252 

weights 82 

Least  squares,  method  of 241-359 

Lengths  of  bars  and  tapes 24,  33 

Leveling : 

barometric 125,  126-130 

geodetic 125-162 

precise  spirit 125,  139-162 

trigonometric 125,  130-139 

Level  work : 

adjustments 160,  344-359 

branch  lines,  circuits  and  nets 359 

closed  circuits 160,  357 

duplicate  lines ' 160,  346 

general  law  of  probable  error 347 

intermediate  points 160,  355 

law  of  relative  weight 348 

level  nets 161,  352 

multiple  lines 160,  350 

probable  error  of  lines  of  unit  length 348,  349 

sectional  lines 347 

simultaneous  lines 160 

Light,  diurnal  aberration  of 213 

L.  M.  Z.  problem 103 

Locating  a  parallel  of  latitude 120 

Locations,  absolute  and  relative 4 

Longitude 109,  197 

astronomical 197 

geodetic 197 

periodic  changes  in 203 

Longitude  determinations 197-203 

at  sea 203 

by  lunar  observations 198 

lunar  culminations 199 

lunar  distances 199 

lunar  occupations 199 

by  special  methods 198 

flash  signals 198 

special  phenomena 198 

by  telegraph 200 

arbitrary  signals 202 


384  INDEX 

PAGE 

by  telegraph — (continued} 

standard  time  signals 201 

star  signals 201 

by  transportation  of  chronometers 199 

Loxodrome 233 

Map  projections 227-240 

conical 234 

Bonne's  projection 238 

Mercator's  conic 236 

simple  conic 235 

cylindrical 229 

Mercator's  cylindrical 231 

rectangular  cylindrical 231 

•    simple  cylindrical 229 

polyconic 240 

rectangular  polyconic 241 

simple  polyconic 240 

trapezoidal , 234 

Mean  absolute  error 305 

Mean  error . 305 

Mean  of  errors 305 

Meaft  radius  of  the  earth 44 

Mean  sea  level 43,  125 

Mean  solar  time 165 

Measures  of  precision 262,  304 

Mercator's  projections: 

conic 236 

cylindrical 231 

Mercurial  barometer 126,  127 

Meridian 167 

lengths 228 

line,  plane,  and  section 167 

Meridians,  convergence  of 88,  111 

Method  of  least  squares 241-359 

Micrometer: 

filar 66 

microscope .  ;  •• 65 

reading  of 67 

run  of 68 

Mistakes 247 

Modulus  of  elasticity 39 

Molitor's  precise  level  rod • .    158 

Most  probable  values  of 

computed  qauntities 296 

conditioned  quantities 2$4-295 

dependent  quantities   284-295 


INDEX  385 

PAGE 

Most  probable  values  of — (continued) 

independent  quantities 266-283 

observed  quantities 242,  266,  295 

Multiple  level  lines 169,  350 

Nadir 167 

Nautical  Almanac 164 

Night  signals 23 

Normal 110 

Normal  equations 273,  276 

law  of  coefficients 280 

Normal  tension 40 

Observation  equations: 

definition  of 271 

reduced 281 

reduction  to  unit  weight 278 

Observations: 

adjustment  of 3,  241-359 

classification  of 243 

Observed  quantities: 

most  probable  values  of 266-295 

probable  errors  of , 297-305 

Observed  values,  definition  of 242 

Ovaloid,  definition  of 105 

Papers  of  U.  S.  Coast  and  Geodetic  Survey 1 

Parallax  (in  altitude) 167,  171 

Parallel  of  latitude,  location  of 120 

Parallels,  length  of  one  degree 228 

Phase 20 

Phaseless  targets 20 

Plane  surveying,  history  of 1 

Plumb-line  deviation 124 

Polar  distance 167 

Pole  signals 20 

Precise  spirit  leveling 125,  139-162 

accuracy  attainable 161 

adjustment  of  results 160,  344-359 

Coast  Survey  precise  level 142,  153 

adjustments  of 155 

constants  of 155 

use  of 156 

European  type  of  precise  level 141,  145 

adjustments  of 146,  150 

constants  of 146 

use  of .  .  152 


386  INDEX 

PAGE 

Precise  spirit  leveling — (continued) 

instruments  used 139,  145,  153 

methods 143,  145 

rods  and  turning  points 158 

sources  of  error 143 

Primary  triangles  and  systems 9 

Prime  vertical .' : 110,  167 

Prime-vertical  transits 192 

Probability: 

equation  of 257,  260 

laws  of  chance 248 

Probable  error: 

general  value  of 29^9 

meaning  of 297 

Probable  errors  of 

angle  measurements 79 

base-line  measurements 46 

computed  quantities 306-311 

conditioned  quantities 304 

dependent  quantities 304 

independent  quantities 300-304 

observed  quantities 297-305 

Projection  of   maps 227-240 

See  Map  projections  for  list  of  types. 

Puissant 's  solution  of  geodetic  problem: 

direct 113 

inverse 118 

Pull,  with  tapes  and  wires 24,  30,  38 

Quadratic  mean 244 

Quadrilateral,  geodetic 7,  90,  327 

algebraic  adjustment  of 90-102 

approximate 92 

rigorous 96 

least  square  adjustment  of 327 

Quantities: 

classification  of 241 

most  probable  values  of 266-296 

computed  quantities 296 

observed  quantities 266-296 

probable  errors  of 297-311 

computed  quantities 306-311 

observed  quantities 297-305 

Radiation,  heat 47 

Reading  micrometers 67 

Reconnoissance .  .  10 


INDEX  387 


Reduced  observation  equations ....*' 281 

Reduction  to  center 75 

Reduction  to  mean  sea  level 43 

Refraction: 

angular 167 

coefficient  of 138 

in  elevation 12 

Relative  locations 4 

Repeating  instruments 47,  49,  52 

adjustments  of 59 

Residual  errors 245 

Residuals 245 

Rhumb  line 233 

Right  ascension 167 

Run  of  micrometer 68 

Sag 24,  30,  39 

Secondary  triangles  and  systems 9 

Sectional  lines: 

base  lines 335 

level  lines 347 

Sidereal  time 165,  168 

Signals  at  stations 18 

board 20 

eccentric 20,  78 

heliotrope 21 

night 23 

phaseless 20 

pole 20 

Simultaneous  level  lines 160 

Single  angle  adjustment 312 

Solar  time 165 

Spherical  excess 88,  89,  90 

Spheroid: 

Bessel's 106 

Clarke's 106 

definition  of 105 

Spirit  leveling,  see  Precise  spirit  leveling. 

Standardizing  bars  and  tapes 33 

Standard  time 165 

Station  adjustment 81,  84,  312,  313-319 

Stations : 

elevation  of 14,  17 

height  of 17 

intervisibility  of 11,  14 

marks 17 

selection  of 10 


388  INDEX 

PAGE 

Stations —  (continued) 

signals  and  targets 18 

towers 17,  18 

triangulation 5 

Steel  tapes 24,  30,  32 

corrections  required  in  tape  measurements 24,  33-39 

standardizing 33 

Steel  and  brass  wires 32 

Systematic  errors 247 

Tables. 361-373 

Tangents 110,  120 

Targets 18 

Telegraphic  determination  of  longitude 200 

Telescope,  zenith 193 

Temperature  corrections  in  base-line  work 24,  31,  36 

Tension,  tapes  and  wires 40 

Tertiary  triangles  and  systems 9 

Theodolite 48 

Theory  of  errors 252-265 

comparison  of  theory  and  experience 264 

Theory  of  weights 81,  243 

Thermometric  base-bars 26 

Tide  gauges: 

automatic 125 

staff 126 

Time 164 

conversion  of 165,  169,  170 

general  principles 164 

varieties  of 165 

Time  determinations 164-186 

at  sea 184 

by  equal  altitudes  of  sun 176 

by  single  altitudes  of  sun 171 

by  sun  and  star  transits 181 

choice  of  methods 184 

Towers,  station  and  signal 17,  18,  47 

Transit,  astronomical 183,  185 

Triangles : 

accuracy  in  closing 102 

adjustment  of 89,  322-326 

classification  of 9 

computation  of 102 

Triangulation : 

adjustments  and  computations 81-102,  312-332 

general  scheme 4 

principles  of 4-23 


INDEX  389 


Triangulation — ( continued} 

stations 5,  10 

systems 5-9 

Trigonometrical  leveling 125,  130-139 

accuracy  attainable .  .  . 139 

observations  at  one  station 133 

reciprocal  observations 136 

sea-horizon  method 131 

Tripods  for 

angle-measuring  instruments 18 

base-bars 27 

leveling  instruments 143 

True  errors 245 

True  values 242 

Turning  points 158 

Uncertainty  of  base-line  measurements 46,  342 

United  States  Coast  and  Geodetic  Survey 1 

papers  of 1 

precise  level 153 

Values,  classification  of 242,  244 

Variations,  periodic: 

in  azimuth 226 

in  latitude : 196 

in  longitude 203 

Vertical  alignment 42 

Weight: 

laws  of 82 

theory  of 81,  243 

Wires,  steel  and  brass 32 

Zenith 167 

Zenith  distance 167 

Zenith  telescope 193 


UNIVERSITY  OF 

OF  CIVIL, 


THIS  BOOK  IS  DUE  ON  THE  LAST  DATE 
STAMPED  BELOW 

AN  INITIAL  FINE  OF  25  CENTS 

WILL  BE  ASSESSED  FOR  FAILURE  TO  RETURN 
THIS  BOOK  ON  THE  DATE  DUE.  THE  PENALTY 
WILL  INCREASE  TO  SO  CENTS  ON  THE  FOURTH 
DAY  AND  TO  $1.OO  ON  THE  SEVENTH  DAY 
OVERDUE. 


JUN 

19  194S 

WAR  2 

4  1947 

... 

f 

X 


C  A  •  .  J  PQrtNlA. 
!L.  ENGINEE 


UNIVERSITY  OF  CALIFORNIA  LIBRARY 


CA 


